let X be ComplexNormSpace; :: thesis: for seq being sequence of X st ( for n being Nat holds seq . n = 0. X ) holds
for m being Nat holds (Partial_Sums ||.seq.||) . m = 0
let seq be sequence of X; :: thesis: ( ( for n being Nat holds seq . n = 0. X ) implies for m being Nat holds (Partial_Sums ||.seq.||) . m = 0 )
assume A1: for n being Nat holds seq . n = 0. X ; :: thesis: for m being Nat holds (Partial_Sums ||.seq.||) . m = 0
let m be Nat; :: thesis: (Partial_Sums ||.seq.||) . m = 0
defpred S₁[ Nat] means ||.seq.|| . $1 = (Partial_Sums ||.seq.||) . $1;
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) = 0 + (||.seq.|| . (k + 1))
.= ||.(0. X).|| + (||.seq.|| . (k + 1))
.= ||.(seq . k).|| + (||.seq.|| . (k + 1)) by A1
.= ((Partial_Sums ||.seq.||) . k) + (||.seq.|| . (k + 1)) by A3, NORMSP_0:def 4
.= (Partial_Sums ||.seq.||) . (k + 1) by SERIES_1:def 1 ; :: thesis: verum

end;

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 NORMSP_0:def 4
.= ||.(0. X).|| by A1
.= 0 ;
:: thesis: verum