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

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume seq ^\ k is summable ; :: thesis: S₁[k + 1]
then Partial_Sums (seq ^\ k) is convergent by Def2;
then A3: (Partial_Sums (seq ^\ k)) ^\ 1 is convergent by CLVECT_2:100;
A4: seq ^\ (k + 1) = (seq ^\ k) ^\ 1 by NAT_1:49;
reconsider seq1 = NAT --> ((seq ^\ k) . 0 ) as sequence of X ;
A5: for m being Element of NAT holds seq1 . m = (seq ^\ k) . 0 by FUNCOP_1:13;
A6: seq1 is convergent by CLVECT_2:1;
Partial_Sums ((seq ^\ k) ^\ 1) = ((Partial_Sums (seq ^\ k)) ^\ 1) - seq1 by A5, Th12;
then Partial_Sums ((seq ^\ k) ^\ 1) is convergent by A3, A6, CLVECT_2:4;
hence S₁[k + 1] by A4, Def2; :: thesis: verum

end;

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