let s be Real_Sequence; :: thesis: ( s is summable implies for n being Element of NAT holds s ^\ n is summable )
defpred S₁[ Element of NAT ] means s ^\ $1 is summable ;
assume s is summable ; :: thesis: for n being Element of NAT holds s ^\ n is summable
then A1: S₁[ 0 ] by NAT_1:48;
A2: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume s ^\ n is summable ; :: thesis: S₁[n + 1]
then Partial_Sums (s ^\ n) is convergent by Def2;
then A3: (Partial_Sums (s ^\ n)) ^\ 1 is convergent ;
A4: s ^\ (n + 1) = (s ^\ n) ^\ 1 by NAT_1:49;
reconsider s1 = NAT --> ((s ^\ n) . 0 ) as Real_Sequence ;
A5: for k being Element of NAT holds s1 . k = (s ^\ n) . 0 by FUNCOP_1:13;
Partial_Sums ((s ^\ n) ^\ 1) = ((Partial_Sums (s ^\ n)) ^\ 1) - s1 by A5, Th14;
then Partial_Sums ((s ^\ n) ^\ 1) is convergent by A3, SEQ_2:25;
hence S₁[n + 1] by A4, Def2; :: thesis: verum

end;

thus for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A1, A2); :: thesis: verum