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

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
reconsider seq1 = NAT --> ((seq ^\ k) . 0) as sequence of X ;
assume seq ^\ k is summable ; :: thesis: S₁[k + 1]
then A2: Partial_Sums (seq ^\ k) is convergent ;
for m being Nat holds seq1 . m = (seq ^\ k) . 0 by ORDINAL1:def 12, FUNCOP_1:7;
then ( seq1 is convergent & Partial_Sums ((seq ^\ k) ^\ 1) = ((Partial_Sums (seq ^\ k)) ^\ 1) - seq1 ) by Th12;
then ( seq ^\ (k + 1) = (seq ^\ k) ^\ 1 & Partial_Sums ((seq ^\ k) ^\ 1) is convergent ) by A2, BHSP_2:4, BHSP_3:31;
hence S₁[k + 1] by Def2; :: thesis: verum

end;

assume seq is summable ; :: thesis: for k being Nat holds seq ^\ k is summable
then A3: S₁[ 0 ] by NAT_1:47;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1); :: thesis: verum