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 Th58:
  seq is summable implies for n holds seq^\n is summable
proof
  assume
A1: seq is summable;
    let n;
A2: Re (seq^\n)=Re seq^\n & Im(seq^\n)=Im seq^\n by Th23;
    Re seq^\n is summable & Im seq^\n is summable by A1,SERIES_1:12;
    hence seq^\n is summable by A2,Th57;
end;
