
theorem copr2:
for F being Field
for p,q being Element of the carrier of Polynom-Ring F
st p,q are_coprime holds p,q have_no_common_roots
proof
let F be Field, p,q be Element of the carrier of Polynom-Ring F;
set P = Polynom-Ring F;
assume AS: p,q are_coprime;
Y: 1.P = 1_.(F) by POLYNOM3:def 10;
now assume p,q have_a_common_root; then
  consider a being Element of F such that
  A: a is_a_common_root_of p,q by RATFUNC1:def 4;
  B: a is_a_root_of p & a is_a_root_of q by A,RATFUNC1:def 3; then
  consider rp being Polynomial of F such that
  C: p = rpoly(1,a) *' rp by HURWITZ:33;
  consider rq being Polynomial of F such that
  D: q = rpoly(1,a) *' rq by B,HURWITZ:33;
  reconsider rpa = rpoly(1,a) as Element of the carrier of P
    by POLYNOM3:def 10;
  rpoly(1,a) divides p & rpoly(1,a) divides q by D,C,RING_4:1; then
  rpa divides 1.P by AS,RING_4:def 10; then
  rpoly(1,a) divides 1_.(F) by Y; then
  deg rpoly(1,a) <= deg 1_.(F) by RING_5:13; then
  1 <= deg 1_.(F) by HURWITZ:27;
  hence contradiction by RATFUNC1:def 2;
  end;
hence thesis;
end;
