ex m being Element of NAT st
for n being Element of NAT st m <= n holds
||.((z rExpSeq ) . n).|| <= (||.z.|| rExpSeq ) . n

proof

take 0 ; :: thesis: for n being Element of NAT st 0 <= n holds
||.((z rExpSeq ) . n).|| <= (||.z.|| rExpSeq ) . n
thus for n being Element of NAT st 0 <= n holds
||.((z rExpSeq ) . n).|| <= (||.z.|| rExpSeq ) . n by Th14; :: thesis: verum

end;

hence z rExpSeq is norm_summable by LOPBAN_3:32, SIN_COS:49; :: thesis: verum