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
  Re (z (#) seq) = Re z (#) Re seq - Im z (#) Im seq & Im (z (#) seq) =
  Re z (#) Im seq + Im z (#) Re seq
proof
  now
    let n be Element of NAT;
    thus Re (z (#) seq).n=Re((z (#) seq).n) by Def5
      .= Re((z * seq.n )) by VALUED_1:6
      .= Re(Re(z) * Re(seq.n) - Im(z) * Im(seq.n)+ (Re(z) * Im(seq.n) + Re(
    seq.n) * Im(z))*<i>) by COMPLEX1:82
      .= Re(z) * Re(seq.n) - Im(z) * Im(seq.n) by COMPLEX1:12
      .= Re(z) *( Re seq.n) - Im(z) * Im(seq.n) by Def5
      .= Re(z) *( Re seq.n) - Im(z) *( Im seq.n) by Def6
      .= (Re(z) (#) Re seq).n - Im(z) *( Im seq.n) by SEQ_1:9
      .= (Re(z) (#) Re seq).n - (Im(z) (#) Im seq).n by SEQ_1:9
      .= (Re(z) (#) Re seq).n + (- (Im(z) (#) Im seq).n)
      .= (Re(z) (#) Re seq).n + (- Im(z) (#) Im seq).n by SEQ_1:10
      .= (Re(z) (#) Re seq - Im(z) (#) Im seq).n by SEQ_1:7;
  end;
  hence Re (z (#) seq)=Re(z)(#)Re seq-Im(z)(#)Im seq;
  now
    let n be Element of NAT;
    thus Im(z(#)seq).n =Im((z (#) seq).n) by Def6
      .=Im((z * seq.n )) by VALUED_1:6
      .=Im(Re(z) * Re(seq.n) - Im(z) * Im(seq.n)+ (Re(z) * Im(seq.n) + Re(
    seq.n) * Im(z))*<i>) by COMPLEX1:82
      .= Re(z) * Im(seq.n) + Re(seq.n) * Im(z) by COMPLEX1:12
      .= Re(z) * Im(seq.n) + Im(z) * Re seq.n by Def5
      .= Re(z) * Im seq.n + Im(z) *(Re seq.n) by Def6
      .= (Re(z) (#) Im seq).n + Im(z) *(Re seq.n) by SEQ_1:9
      .= (Re(z) (#) Im seq).n + (Im(z) (#) Re seq).n by SEQ_1:9
      .= (Re(z) (#) Im seq + Im(z) (#) Re seq).n by SEQ_1:7;
  end;
  hence thesis;
end;
