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 Th15:
  Re seq1 + Re seq2 = Re (seq1+seq2) & Im seq1 + Im seq2 = Im (
  seq1+seq2 qua Complex_Sequence)
proof
  now
    let n be Element of NAT;
    thus (Re seq1+Re seq2).n=Re seq1.n+Re seq2.n by SEQ_1:7
      .=Re(seq1.n)+Re seq2.n by Def5
      .=Re(seq1.n)+Re(seq2.n) by Def5
      .=Re(seq1.n+seq2.n) by COMPLEX1:8
      .=Re((seq1+seq2).n) by VALUED_1:1
      .=Re (seq1+seq2).n by Def5;
  end;
  hence Re seq1+Re seq2=Re (seq1+seq2);
  now
    let n be Element of NAT;
    thus (Im seq1+Im seq2).n=Im seq1.n+Im seq2.n by SEQ_1:7
      .=Im(seq1.n)+Im seq2.n by Def6
      .=Im(seq1.n)+Im(seq2.n) by Def6
      .=Im(seq1.n+seq2.n) by COMPLEX1:8
      .=Im((seq1+seq2).n) by VALUED_1:1
      .=Im (seq1+seq2).n by Def6;
  end;
  hence thesis;
end;
