COMSEQ

:: COMSEQ_3 semantic presentation

begin

theorem :: COMSEQ_3:2

theorem :: COMSEQ_3:3

registration

cluster Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) summable -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

registration

cluster Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) absolutely_summable -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

registration

cluster Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued absolutely_summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: COMSEQ_3:4

theorem :: COMSEQ_3:5

theorem :: COMSEQ_3:6

theorem :: COMSEQ_3:7

theorem :: COMSEQ_3:8

for rseq1, rseq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) )
for p being ( ( ) ( complex real ext-real ) Real) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) <= p : ( ( ) ( complex real ext-real ) Real) * (rseq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) holds
for n, l being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + l : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((Partial_Sums rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + l : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) - ((Partial_Sums rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) <= p : ( ( ) ( complex real ext-real ) Real) * (((Partial_Sums rseq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + l : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) - ((Partial_Sums rseq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ;

theorem :: COMSEQ_3:9

for rseq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V48() V61() V62() V63() V64() V65() V66() V67() V68() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) <= rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) holds
( ( for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) - ((Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = ((Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) - ((Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds abs ((Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = (Partial_Sums rseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ) ;

theorem :: COMSEQ_3:10

begin

definition

let z be ( ( complex ) ( complex ) number ) ;

func z GeoSeq -> ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) means :: COMSEQ_3:def 1
( it : ( ( non zero ) ( non zero ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V48() V61() V62() V63() V64() V65() V66() V67() V68() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( non zero ) ( non zero ) set ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = (it : ( ( non zero ) ( non zero ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) * z : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) );

end;

notation

let z be ( ( complex ) ( complex ) number ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) number ) ;
synonym z #N n for z |^ n;

end;

definition

let z be ( ( complex ) ( complex ) number ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) number ) ;

:: original: #N
redefine func z #N n -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) equals :: COMSEQ_3:def 2
(z : ( ( ) ( ) set ) GeoSeq) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( non zero ) ( non zero ) set ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ;

end;

theorem :: COMSEQ_3:11

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) #N 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V48() V61() V62() V63() V64() V65() V66() V67() V68() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ;

definition

let f be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;

func Re f -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: COMSEQ_3:def 3
( dom it : ( ( non zero ) ( non zero ) set ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( non zero ) ( non zero ) set ) : ( ( ) ( ) set ) holds
it : ( ( non zero ) ( non zero ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = Re (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) );

func Im f -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: COMSEQ_3:def 4
( dom it : ( ( non zero ) ( non zero ) set ) : ( ( ) ( ) set ) = dom f : ( ( ) ( ) set ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom it : ( ( non zero ) ( non zero ) set ) : ( ( ) ( ) set ) holds
it : ( ( non zero ) ( non zero ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) = Im (f : ( ( ) ( ) set ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) );

end;

registration

let f be ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) Function) ;

cluster Re f : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like real-valued ;

cluster Im f : ( ( Relation-like Function-like complex-valued ) ( Relation-like Function-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) -> Relation-like Function-like real-valued ;

end;

definition

let X be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like complex-valued ) PartFunc of ,) ;
:: original: Re
redefine func Re f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
:: original: Im
redefine func Im f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;

end;

definition

let c be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

:: original: Re
redefine func Re c -> ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) means :: COMSEQ_3:def 5
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( non zero ) ( non zero ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Re (c : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ;

:: original: Im
redefine func Im c -> ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) means :: COMSEQ_3:def 6
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( non zero ) ( non zero ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Im (c : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ;

end;

theorem :: COMSEQ_3:12

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) <= (abs (Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) + (abs (Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ;

theorem :: COMSEQ_3:13

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds
( abs (Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) <= |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & abs (Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) <= |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ;

theorem :: COMSEQ_3:14

theorem :: COMSEQ_3:15

theorem :: COMSEQ_3:16

theorem :: COMSEQ_3:17

for r being ( ( ) ( complex real ext-real ) Real)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds
( r : ( ( ) ( complex real ext-real ) Real) * (Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Re (r : ( ( ) ( complex real ext-real ) Real) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & r : ( ( ) ( complex real ext-real ) Real) * (Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Im (r : ( ( ) ( complex real ext-real ) Real) * z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ;

theorem :: COMSEQ_3:18

theorem :: COMSEQ_3:19

theorem :: COMSEQ_3:20

for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence)
for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds
( Re (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = ((Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) (#) (Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) (#) (Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & Im (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = ((Re z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) (#) (Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + ((Im z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) (#) (Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMSEQ_3:21

for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) holds
( Re (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = ((Re seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (Re seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((Im seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (Im seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & Im (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = ((Re seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (Im seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + ((Im seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) (Re seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

definition

let Nseq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let seq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;
:: original: (#)
redefine func seq * Nseq -> ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

theorem :: COMSEQ_3:22

for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence)
for Nseq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( Re (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) * Nseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = (Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * Nseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Re b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) & Im (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) * Nseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = (Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) * Nseq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) increasing ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued natural-valued increasing non-decreasing ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Im b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) ) ;

theorem :: COMSEQ_3:23

definition

let s be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;
:: original: Partial_Sums
redefine func Partial_Sums s -> ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

definition

func Sum seq -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) equals :: COMSEQ_3:def 7
lim (Partial_Sums seq : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ;

end;

theorem :: COMSEQ_3:24

theorem :: COMSEQ_3:25

theorem :: COMSEQ_3:26

theorem :: COMSEQ_3:27

theorem :: COMSEQ_3:28

theorem :: COMSEQ_3:29

theorem :: COMSEQ_3:30

theorem :: COMSEQ_3:31

theorem :: COMSEQ_3:32

for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence)
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Partial_Sums (Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Partial_Sums (Re b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) = Re ((Partial_Sums seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) subsequence of Partial_Sums b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) & (Partial_Sums (Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Partial_Sums (Im b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) = Im ((Partial_Sums seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) subsequence of Partial_Sums b₁ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) ;

theorem :: COMSEQ_3:33

for seq1, seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V48() V61() V62() V63() V64() V65() V66() V67() V68() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) holds
Partial_Sums (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ^\ 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) subsequence of b₂ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) = ((Partial_Sums seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ^\ 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) subsequence of Partial_Sums b₂ : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) - seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_3:34

registration

cluster Partial_Sums |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ext-real-valued real-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) non-decreasing for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

end;

theorem :: COMSEQ_3:35

theorem :: COMSEQ_3:36

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) st 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) <> z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) GeoSeq) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = (1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) - (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) #N (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) / (1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) - z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ;

theorem :: COMSEQ_3:37

begin

theorem :: COMSEQ_3:38

for a, b being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for c being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( Re (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & Im (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ) holds
( ( a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent ) iff c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is convergent ) ;

theorem :: COMSEQ_3:39

for a, b being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence)
for c being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( Re (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & Im (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ) holds
( c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is convergent & lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = (lim a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) + ((lim b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) ) ;

theorem :: COMSEQ_3:40

for a, b being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for c being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( Re (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & Im (c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ) holds
( a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim a : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Re (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & lim b : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Im (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ;

theorem :: COMSEQ_3:41

for c being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds
( Re c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & Im c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim (Re c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Re (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & lim (Im c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = Im (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ;

registration

let c be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ;

cluster Re c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

cluster Im c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) convergent for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

end;

theorem :: COMSEQ_3:42

for c being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st Re c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & Im c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
( c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is convergent & Re (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = lim (Re c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) & Im (lim c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = lim (Im c : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ;

theorem :: COMSEQ_3:43

theorem :: COMSEQ_3:44

theorem :: COMSEQ_3:45

theorem :: COMSEQ_3:46

theorem :: COMSEQ_3:47

theorem :: COMSEQ_3:48

begin

theorem :: COMSEQ_3:49

for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) subsequence of seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ) holds
( Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Re seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) & Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Im seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) ) ;

theorem :: COMSEQ_3:50

definition

attr seq is summable means :: COMSEQ_3:def 8
Partial_Sums seq : ( ( ) ( ) set ) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is convergent ;

end;

registration

cluster Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

registration

let seq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued summable ) Complex_Sequence) ;

cluster Partial_Sums seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

definition

attr seq is absolutely_summable means :: COMSEQ_3:def 9
|.seq : ( ( ) ( ) set ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable ;

end;

theorem :: COMSEQ_3:51

registration

cluster Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

registration

let seq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) absolutely_summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable ) Complex_Sequence) ;

cluster |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) absolutely_summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Relation-like Function-like real-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ext-real-valued real-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

end;

theorem :: COMSEQ_3:52

registration

cluster Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) convergent for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: COMSEQ_3:53

registration

cluster Re seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

cluster Im seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

end;

theorem :: COMSEQ_3:54

theorem :: COMSEQ_3:55

registration

let seq1, seq2 be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Complex_Sequence) ;

cluster seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

cluster seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

theorem :: COMSEQ_3:56

registration

let z be ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ;
let seq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Complex_Sequence) ;

cluster z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

theorem :: COMSEQ_3:57

theorem :: COMSEQ_3:58

registration

cluster seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued bounded convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like total complex-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) ;

end;

theorem :: COMSEQ_3:59

theorem :: COMSEQ_3:60

theorem :: COMSEQ_3:61

registration

let seq be ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) absolutely_summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable ) Complex_Sequence) ;

cluster Partial_Sums |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) absolutely_summable ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued convergent summable ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like total complex-valued ext-real-valued real-valued ) set ) -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) bounded_above for ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

end;

theorem :: COMSEQ_3:62

theorem :: COMSEQ_3:63

registration

cluster Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) absolutely_summable -> Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

registration

cluster Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued absolutely_summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: COMSEQ_3:64

for z being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) st |.z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) < 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) GeoSeq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is summable & Sum (z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) GeoSeq) : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) = 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) / (1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) - z : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ;

theorem :: COMSEQ_3:65

theorem :: COMSEQ_3:66

theorem :: COMSEQ_3:67

theorem :: COMSEQ_3:68

theorem :: COMSEQ_3:69

theorem :: COMSEQ_3:70

for rseq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) -root (|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) & ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
not |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable ;

theorem :: COMSEQ_3:71

theorem :: COMSEQ_3:72

for rseq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-increasing & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) * (|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) holds
( seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is absolutely_summable iff rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ) ;

theorem :: COMSEQ_3:73

theorem :: COMSEQ_3:74

theorem :: COMSEQ_3:75

for rseq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) <> 0c : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative V62() V63() V64() V65() V66() V67() V68() ) Element of COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) & rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) = (|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) / (|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ) & rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim rseq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) < 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative V48() V61() V62() V63() V64() V65() V66() V67() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) ) holds
seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) -valued Function-like non zero total V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V62() V63() V64() V65() V66() V67() V68() ) Element of K19(REAL : ( ( ) ( non zero V50() V62() V63() V64() V68() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V50() V62() V68() ) set ) ) complex-valued ) Complex_Sequence) is absolutely_summable ;

theorem :: COMSEQ_3:76