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
  seq is subsequence of seq1 implies Re seq is subsequence of Re seq1 &
  Im seq is subsequence of Im seq1
proof
  assume seq is subsequence of seq1;
  then consider Nseq such that
A1: seq=seq1*Nseq by VALUED_0:def 17;
  Re seq=Re seq1* Nseq by A1,Th22;
  hence Re seq is subsequence of Re seq1;
  Im seq=Im seq1 * Nseq by A1,Th22;
  hence thesis;
end;
