let X be RealNormSpace; :: thesis: for s being sequence of X st s is summable holds
for n being Nat holds s ^\ n is summable
let s be sequence of X; :: thesis: ( s is summable implies for n being Nat holds s ^\ n is summable )
defpred S₁[ Nat] means s ^\ $1 is summable ;
A1: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
reconsider s1 = NAT --> ((s ^\ n) . 0) as sequence of X ;
for k being Nat holds s1 . k = (s ^\ n) . 0 by ORDINAL1:def 12, FUNCOP_1:7;
then A2: Partial_Sums ((s ^\ n) ^\ 1) = ((Partial_Sums (s ^\ n)) ^\ 1) - s1 by Th21;
assume s ^\ n is summable ; :: thesis: S₁[n + 1]
then Partial_Sums (s ^\ n) is convergent ;
then A3: (Partial_Sums (s ^\ n)) ^\ 1 is convergent by Th7;
s1 is convergent by Th12;
then ( s ^\ (n + 1) = (s ^\ n) ^\ 1 & Partial_Sums ((s ^\ n) ^\ 1) is convergent ) by A3, A2, NAT_1:48, NORMSP_1:20;
hence S₁[n + 1] by Def1; :: thesis: verum

end;

assume s is summable ; :: thesis: for n being Nat holds s ^\ n is summable
then A4: S₁[ 0 ] by NAT_1:47;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A1); :: thesis: verum