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;
