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 S_{1}[ Nat] means s ^\ $1 is summable ;

A1: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]

then A4: S_{1}[ 0 ]
by NAT_1:47;

thus for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A4, A1); :: thesis: verum

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 S

A1: for n being Nat st S

S

proof

assume
s is summable
; :: thesis: for n being Nat holds s ^\ n is summable
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[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: S_{1}[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 S_{1}[n + 1]
by Def1; :: thesis: verum

end;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: S

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 S

then A4: S

thus for n being Nat holds S