let X be ComplexNormSpace; :: thesis: for seq being sequence of X st ( for n being Element of NAT holds seq . n = 0. X ) holds
for m being Element of NAT holds (Partial_Sums ||.seq.||) . m = 0
let seq be sequence of X; :: thesis: ( ( for n being Element of NAT holds seq . n = 0. X ) implies for m being Element of NAT holds (Partial_Sums ||.seq.||) . m = 0 )
assume A1: for n being Element of NAT holds seq . n = 0. X ; :: thesis: for m being Element of NAT holds (Partial_Sums ||.seq.||) . m = 0
let m be Element of NAT ; :: thesis: (Partial_Sums ||.seq.||) . m = 0
defpred S₁[ Element of NAT ] means ||.seq.|| . $1 = (Partial_Sums ||.seq.||) . $1;
A2: S₁[ 0 ] by SERIES_1:def 1;
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: S₁[k] ; :: thesis: S₁[k + 1]
thus ||.seq.|| . (k + 1) = 0 + (||.seq.|| . (k + 1))
.= ||.(0. X).|| + (||.seq.|| . (k + 1)) by CLVECT_1:103
.= ||.(seq . k).|| + (||.seq.|| . (k + 1)) by A1
.= ((Partial_Sums ||.seq.||) . k) + (||.seq.|| . (k + 1)) by A4, CLVECT_1:def 17
.= (Partial_Sums ||.seq.||) . (k + 1) by SERIES_1:def 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence (Partial_Sums ||.seq.||) . m = ||.seq.|| . m
.= ||.(seq . m).|| by CLVECT_1:def 17
.= ||.(0. X).|| by A1
.= 0 by CLVECT_1:103 ;
:: thesis: verum