
theorem mm6:
for F being 0-characteristic Field
for p being irreducible Element of the carrier of Polynom-Ring F
holds p gcd (Deriv F).p = 1_.(F)
proof
let F be 0-characteristic Field,
    p be irreducible Element of the carrier of Polynom-Ring F;
reconsider e = 1_.(F) as Element of Polynom-Ring F by POLYNOM3:def 10;
reconsider q1 = p, q2 = (Deriv F).p as Element of Polynom-Ring F;
H: q1 <> 0_.(F) &
   e is Element of the carrier of Polynom-Ring F;
   e * q1 = (1_.(F)) *' p by POLYNOM3:def 10 .= q1; then
A: e divides q1 by GCD_1:def 1;
   e * q2 = (1_.(F)) *' ((Deriv F).p) by POLYNOM3:def 10 .= q2; then
B: e divides q2 by GCD_1:def 1;
now let r1 be Element of Polynom-Ring F;
  assume C1: r1 divides q1 & r1 divides q2;
  reconsider r = r1 as Element of the carrier of Polynom-Ring F;
  consider u1 being Element of Polynom-Ring F such that
  C2: q1 = r1 * u1 by C1,GCD_1:def 1;
  reconsider u = u1 as Element of the carrier of Polynom-Ring F;
  C3: p = r *' u by C2,POLYNOM3:def 10;
  consider u2 being Element of Polynom-Ring F such that
  C5: q2 = r1 * u2 by C1,GCD_1:def 1;
  reconsider v = u2 as Element of the carrier of Polynom-Ring F;
  C7: (Deriv F).p = r *' v by C5,POLYNOM3:def 10;
  per cases by C3,RING_4:1,RING_4:41;
  suppose deg r < 1 + 0; then
    r is constant by INT_1:7,RING_4:def 4; then
    consider a being Element of F such that D1: r = a|F by RING_4:20;
    D2: a <> 0.F by C3,D1;
    reconsider v = (a")|F as Element of Polynom-Ring F by POLYNOM3:def 10;
    r1 * v = r *' ((a")|F) by POLYNOM3:def 10
          .= (a * (a"))|F by D1,RING_4:18
          .= (1.F)|F by D2,VECTSP_1:def 10
          .= 1_.F by RING_4:14;
    hence r1 divides e by GCD_1:def 1;
    end;
  suppose D1: deg r >= deg p;
    deg r <= deg p by C3,RING_4:1,RING_5:13; then
    D2: deg r = deg p by D1,XXREAL_0:1;
    (Deriv F).p <> 0_.(F) by mm6b; then
    (Deriv F).p is non zero by UPROOTS:def 5; then
    deg r <= deg (Deriv F).p by C7,RING_4:1,RING_5:13;
    hence r1 divides e by D2,mm6a;
    end;
  end;
then e is q1,q2-gcd by A,B,RING_4:def 10;
then 1_.F = q1 gcd q2 by H,RING_4:def 11;
hence thesis by RING_4:def 12;
end;
