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 Th32:
  Partial_Sums(Re seq)^\k =Re (Partial_Sums(seq)^\k) &
  Partial_Sums(Im seq)^\k =Im (Partial_Sums(seq)^\k)
proof
  now
    let n be Element of NAT;
    thus (Partial_Sums(Re seq)^\k).n =Partial_Sums(Re seq).(n+k) by NAT_1:def 3
      .=Re Partial_Sums(seq).(n+k) by Th26
      .=Re(Partial_Sums(seq).(n+k)) by Def5
      .=Re((Partial_Sums(seq)^\k).n) by NAT_1:def 3
      .=(Re (Partial_Sums(seq)^\k)).n by Def5;
    thus (Partial_Sums(Im seq)^\k).n =Partial_Sums(Im seq).(n+k) by NAT_1:def 3
      .=Im Partial_Sums(seq).(n+k) by Th26
      .=Im(Partial_Sums(seq).(n+k)) by Def6
      .=Im((Partial_Sums(seq)^\k).n) by NAT_1:def 3
      .=Im (Partial_Sums(seq)^\k).n by Def6;
  end;
  hence thesis;
end;
