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 Th16:
  -(Re seq) = Re (-seq) & -(Im seq) = Im -seq
proof
  now
    let n be Element of NAT;
    thus (-Re seq).n= -(Re seq.n) by SEQ_1:10
      .=-Re(seq.n) by Def5
      .= Re(-(seq.n)) by COMPLEX1:17
      .= Re((-seq).n) by VALUED_1:8
      .=Re (-seq).n by Def5;
  end;
  hence -Re seq=Re (-seq);
  now
    let n be Element of NAT;
    thus (-Im seq).n=-Im seq.n by SEQ_1:10
      .=-Im(seq.n) by Def6
      .= Im(-(seq.n)) by COMPLEX1:17
      .= Im((-seq).n) by VALUED_1:8
      .=(Im -seq).n by Def6;
  end;
  hence thesis;
end;
