let seq be Complex_Sequence; :: thesis: ( ( for n being Element of NAT holds seq . n = 0c ) implies ( seq is summable & Sum seq = 0c ) )
assume A1: for n being Element of NAT holds seq . n = 0c ; :: thesis: ( seq is summable & Sum seq = 0c )
A2: for m being Element of NAT holds (Partial_Sums seq) . m = 0c

proof

defpred S₁[ Element of NAT ] means seq . $1 = (Partial_Sums seq) . $1;
let m be Element of NAT ; :: thesis: (Partial_Sums seq) . m = 0c
A3: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: seq . k = (Partial_Sums seq) . k ; :: thesis: S₁[k + 1]
thus seq . (k + 1) = 0c + (seq . (k + 1))
.= (seq . k) + (seq . (k + 1)) by A1
.= (Partial_Sums seq) . (k + 1) by A4, SERIES_1:def 1 ; :: thesis: verum

end;

A5: S₁[ 0 ] by SERIES_1:def 1;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A5, A3);
hence (Partial_Sums seq) . m = seq . m
.= 0c by A1 ;
:: thesis: verum

end;

A6: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((Partial_Sums seq) . m) - 0c).| < p

proof

let p be Real; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((Partial_Sums seq) . m) - 0c).| < p )
assume A7: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.(((Partial_Sums seq) . m) - 0c).| < p
take 0 ; :: thesis: for m being Element of NAT st 0 <= m holds
|.(((Partial_Sums seq) . m) - 0c).| < p
let m be Element of NAT ; :: thesis: ( 0 <= m implies |.(((Partial_Sums seq) . m) - 0c).| < p )
assume 0 <= m ; :: thesis: |.(((Partial_Sums seq) . m) - 0c).| < p
thus |.(((Partial_Sums seq) . m) - 0c).| < p by A2, A7, COMPLEX1:44; :: thesis: verum

end;

then A8: Partial_Sums seq is convergent by COMSEQ_2:def 4;
then lim (Partial_Sums seq) = 0c by A6, COMSEQ_2:def 5;
hence ( seq is summable & Sum seq = 0c ) by A8, COMSEQ_3:def 7, COMSEQ_3:def 8; :: thesis: verum