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 Th42:
  for c being Complex_Sequence st Re c is convergent & Im c is
convergent holds c is convergent & Re(lim(c))=lim(Re c) & Im (lim(c))=lim(Im c)
proof
  let c be Complex_Sequence;
  assume
A1: Re c is convergent & Im c is convergent;
A2: ( for n holds Re c.n=Re(c.n))& for n holds Im c.n=Im(c.n) by Def5,Def6;
  then lim(c) = lim(Re c)+(lim(Im c))*<i> by A1,Th39;
  hence thesis by A1,A2,Th39,COMPLEX1:12;
end;
