
theorem
for p being Hurwitz Polynomial of F_Complex
holds even_part(p), odd_part(p) have_no_common_roots
proof
let p be Hurwitz Polynomial of F_Complex;
set e = even_part(p), o = odd_part(p);
let x be Element of F_Complex;
  assume A1: x is_a_common_root_of e,o;
  A2: x is_a_root_of e+o by A1,RATFUNC1:16;
  e + o = p by Th9;
  then A3: Re(x) < 0 & Re(-x) < 0 by A1,Th26,A2,HURWITZ:def 8;
  reconsider s = x as Complex;
  Re(-s) = - Re(s) by COMPLEX1:17;
  hence contradiction by A3,COMPLFLD:2;
end;
