reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th15:
  for R being real Polynomial of F_Complex
    for r being real Element of F_Complex holds
      eval(R,r) is real
proof
  let I be real Polynomial of FC;
  let r be real Element of FC;
  consider H be FinSequence of FC such that
  A1:eval(I,r) = Sum H & len H = len I and
  A2:  for n be Element of NAT st n in dom H holds H.n =
  I.(n-'1) * (power FC).(r,n-'1) by POLYNOM4:def 2;
  consider h being sequence of the carrier of FC such that
  A3:Sum H = h.(len H) and
  A4:h.0 = 0.FC & for j being Nat for v being Element of FC st
    j < len H & v = H.(j + 1) holds h.(j + 1) = h.j + v
  by RLVECT_1:def 12;
  defpred P[Nat] means $1 <= len H implies h.$1 is real;
  A5:P[0] by A4;
  A6:P[n] implies P[n+1]
  proof
    assume A7:P[n];set n1=n+1;
    assume A8:n1 <= len H;
    n1 in dom H by NAT_1:11,A8,FINSEQ_3:25;
    then A9: H.n1 =  I.(n1-'1) * (power FC).(r,n1-'1) by A2;
    reconsider Hn1=H.n1 as real Element of FC by A9;
    h.n1 = h.n + Hn1 by A4,A8,NAT_1:13;
    hence thesis by A8,A7,NAT_1:13;
  end;
  P[n] from NAT_1:sch 2(A5,A6);
  hence thesis by A1,A3;
end;
