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 Th40:
  for a, b being Real_Sequence, c being convergent
  Complex_Sequence st (for n holds Re (c.n) = a.n & Im (c.n) = b.n) holds a is
  convergent & b is convergent & lim a=Re lim c & lim b=Im lim c
proof
  let a, b be Real_Sequence, c be convergent Complex_Sequence;
  assume
A1: for n holds Re(c.n) = a.n & Im(c.n) = b.n;
A2: for p being Real st 0<p ex n st for m st n<=m holds |.b.m-Im(
  lim(c)).| <p
  proof
    let p be Real;
    assume
A3: p>0;
    consider n such that
A4: for m st n <= m holds |.c.m-lim(c).| < p by A3,COMSEQ_2:def 6;
    take n;
    let m;
    assume n <= m;
    then
A5: |.c.m-lim(c).| < p by A4;
    Im(c.m) =b.m & Im(c.m-lim(c))=Im(c.m)-Im(lim(c)) by A1,COMPLEX1:19;
    then |.b.m-Im(lim(c)) .| <= |.c.m-lim(c).| by Th13;
    hence thesis by A5,XXREAL_0:2;
  end;
A6: for p being Real st 0<p ex n st for m st n<=m holds |.a.m-Re(
  lim(c)).|<p
  proof
    let p be Real;
    assume
A7: 0<p;
    consider n such that
A8: for m st n <= m holds |.c.m-lim(c).| < p by A7,COMSEQ_2:def 6;
    take n;
    let m;
    Re(c.m) =a.m by A1;
    then Re(c.m-lim(c))=a.m-Re(lim(c)) by COMPLEX1:19;
    then
A9: |.a.m-Re(lim(c)) .| <= |.c.m-lim(c).| by Th13;
    assume n <= m;
    then |.c.m-lim(c).| < p by A8;
    hence thesis by A9,XXREAL_0:2;
  end;
  a is convergent & b is convergent by A1,Th38;
  hence thesis by A6,A2,SEQ_2:def 7;
end;
