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 Th39:
  for a, b being convergent Real_Sequence, c being
  Complex_Sequence st (for n holds Re (c.n) = a.n & Im (c.n) = b.n) holds c is
  convergent & lim(c)= lim(a) + lim(b)*<i>
proof
  let a, b be convergent Real_Sequence, c be Complex_Sequence;
  reconsider g=lim(a)+lim(b)*<i> as Element of COMPLEX by XCMPLX_0:def 2;
  assume
A1: for n holds Re(c.n) = a.n & Im(c.n) = b.n;
  hence
A2: c is convergent by Th38;
  for p being Real st 0<p ex n st for m st n<=m holds |.c.m-g.| < p
  proof
    let p be Real;
    assume
A3: 0 < p;
    then consider n1 such that
A4: for m st n1<=m holds |.a.m-lim(a).|< p/2 by SEQ_2:def 7;
    consider n2 such that
A5: for m st n2<=m holds |.b.m-lim(b).|< p/2 by A3,SEQ_2:def 7;
    n1 <= n1+n2 & n2 <= n1+n2 by NAT_1:11;
    then consider n such that
A6: n1 <= n and
A7: n2 <= n;
    take n;
    let m;
    assume
A8: n <= m;
    then n2 <= m by A7,XXREAL_0:2;
    then
A9: |.b.m-lim(b).|< p/2 by A5;
    n1 <= m by A6,A8,XXREAL_0:2;
    then |.a.m-lim(a).|< p/2 by A4;
    then
A10: |.a.m-lim(a).| + |.b.m-lim(b).|< p/2+ p/2 by A9,XREAL_1:8;
    Re(c.m)=a.m & Re(g) =lim(a) by A1,COMPLEX1:12;
    then
A11: Re(c.m-g)=a.m-lim(a) by COMPLEX1:19;
    Im(c.m)=b.m & Im(g) =lim(b) by A1,COMPLEX1:12;
    then
A12: Im(c.m-g)=b.m-lim(b) by COMPLEX1:19;
    |.c.m-g.| <= |.Re(c.m-g).| + |.Im(c.m-g).| by Th12;
    hence thesis by A10,A11,A12,XXREAL_0:2;
  end;
  hence thesis by A2,COMSEQ_2:def 6;
end;
