
theorem mm4i:
for F being Field
for p being non zero Element of the carrier of Polynom-Ring F
holds p gcd (Deriv F).p = 1_.(F) implies p is square-free
proof
let F be Field;
let p be non zero Element of the carrier of Polynom-Ring F;
now assume not p is square-free; then
   consider q1 being non constant Polynomial of F such that
   A1: q1`^2 divides p by lemsq;
   consider r1 being Polynomial of F such that
   A2: p = (q1`^2) *' r1 by A1,RING_4:1;
   reconsider r = r1, q = q1 as Element of the carrier of Polynom-Ring F
      by POLYNOM3:def 10;
   q1`^2 = q1 *' q1 by POLYNOM5:17 .= q * q by POLYNOM3:def 10; then
   A3: p = (q * q) * r by A2,POLYNOM3:def 10 .= q * (q * r) by GROUP_1:def 3;
   A4: p = (q1 *' q1) *' r1 by A2,POLYNOM5:17
        .= q1 *' (q1 *' r) by POLYNOM3:33;
   reconsider u = (Deriv F).(q*r) + r * ((Deriv F).q) as Polynomial of F
     by POLYNOM3:def 10;
   (Deriv F).p
      = q * ((Deriv F).(q*r)) + (q*r) * ((Deriv F).q) by A3,RINGDER1:def 1
     .= q * ((Deriv F).(q*r)) + q * (r * ((Deriv F).q)) by GROUP_1:def 3
     .= q * ((Deriv F).(q*r) + r * ((Deriv F).q)) by VECTSP_1:def 2
     .= q1 *' u by POLYNOM3:def 10; then
   A5: q1 divides p & q1 divides (Deriv F).p by A4,RING_4:1;
   deg q1 > 0 & deg 1_.(F) <= 0 by RATFUNC1:def 2;
   hence p gcd (Deriv F).p <> 1_.(F) by A5,RING_5:13,RING_4:52;
   end;
hence thesis;
end;
