defpred S₁[ set ] means (Partial_Sums |.(0c ExpSeq).|) . $1 = 1;
(Partial_Sums |.(0c ExpSeq).|) . 0 = |.(0c ExpSeq).| . 0 by SERIES_1:def 1
.= |.((0c ExpSeq) . 0).| by VALUED_1:18
.= 1 by Th4, COMPLEX1:48 ;
then A1: S₁[ 0 ] ;
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: (Partial_Sums |.(0c ExpSeq).|) . n = 1 ; :: thesis: S₁[n + 1]
A4: n in NAT by ORDINAL1:def 12;
hence (Partial_Sums |.(0c ExpSeq).|) . (n + 1) = 1 + (|.(0c ExpSeq).| . (n + 1)) by A3, SERIES_1:def 1
.= 1 + |.((0c ExpSeq) . (n + 1)).| by VALUED_1:18
.= 1 + |.((((0c ExpSeq) . n) * 0c) / ((n + 1) + (0 * <i>))).| by A4, Th4
.= 1 by COMPLEX1:44 ;
:: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
then Partial_Sums |.(0c ExpSeq).| is constant by VALUED_0:def 18;
then A5: |.(0c ExpSeq).| is summable by SERIES_1:def 2;
defpred S₂[ set ] means (Partial_Sums (0c ExpSeq)) . $1 = 1;
(Partial_Sums (0c ExpSeq)) . 0 = (0c ExpSeq) . 0 by SERIES_1:def 1
.= 1 by Th4 ;
then A6: S₂[ 0 ] ;
A7: for n being Element of NAT st S₂[n] holds
S₂[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₂[n] implies S₂[n + 1] )
assume A8: (Partial_Sums (0c ExpSeq)) . n = 1 ; :: thesis: S₂[n + 1]
thus (Partial_Sums (0c ExpSeq)) . (n + 1) = 1r + ((0c ExpSeq) . (n + 1)) by A8, SERIES_1:def 1
.= 1r + ((((0c ExpSeq) . n) * 0c) / ((n + 1) + (0 * <i>))) by Th4
.= 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₂[n] from NAT_1:sch 1(A6, A7);
hence ( 0c ExpSeq is absolutely_summable & Sum (0c ExpSeq) = 1r ) by A5, COMSEQ_2:10, COMSEQ_3:def 9; :: thesis: verum