let seq be Complex_Sequence; :: thesis: ( ( for n being Nat holds seq . n = 0c ) implies for m being Nat holds (Partial_Sums |.seq.|) . m = 0 )
defpred S₁[ Nat] means |.seq.| . $1 = (Partial_Sums |.seq.|) . $1;
assume A1: for n being Nat holds seq . n = 0c ; :: thesis: for m being Nat holds (Partial_Sums |.seq.|) . m = 0
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
thus |.seq.| . (k + 1) = |.0c.| + (|.seq.| . (k + 1)) by COMPLEX1:44
.= |.(seq . k).| + (|.seq.| . (k + 1)) by A1
.= ((Partial_Sums |.seq.|) . k) + (|.seq.| . (k + 1)) by A3, VALUED_1:18
.= (Partial_Sums |.seq.|) . (k + 1) by SERIES_1:def 1 ; :: thesis: verum

end;

let m be Nat; :: thesis: (Partial_Sums |.seq.|) . m = 0
A4: S₁[ 0 ] by SERIES_1:def 1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2);
hence (Partial_Sums |.seq.|) . m = |.seq.| . m
.= |.(seq . m).| by VALUED_1:18
.= 0 by A1, COMPLEX1:44 ;
:: thesis: verum