let s be Real_Sequence; :: 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] )
set s1 = seq_const ((s ^\ n) . 0);
for k being Nat holds (seq_const ((s ^\ n) . 0)) . k = (s ^\ n) . 0 by SEQ_1:57;
then A2: Partial_Sums ((s ^\ n) ^\ 1) = ((Partial_Sums (s ^\ n)) ^\ 1) - (seq_const ((s ^\ n) . 0)) by Th11;
assume s ^\ n is summable ; :: thesis: S₁[n + 1]
then Partial_Sums (s ^\ n) is convergent ;
then ( s ^\ (n + 1) = (s ^\ n) ^\ 1 & Partial_Sums ((s ^\ n) ^\ 1) is convergent ) by A2, NAT_1:48;
hence S₁[n + 1] by Def2; :: thesis: verum

end;

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