reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem
  (ex n st seq^\n is summable) implies seq is summable
proof
  given n such that
A1: seq^\n is summable;
  Im (seq^\n)=Im seq^\n by Th23;
  then
A2: Im seq is summable by A1,SERIES_1:13;
  Re (seq^\n)=Re seq^\n by Th23;
  then Re seq is summable by A1,SERIES_1:13;
  hence thesis by A2,Th57;
end;
