
theorem mm6c:
for F being 0-characteristic Field
for p being non zero Element of the carrier of Polynom-Ring F
holds deg (Deriv F).p = deg p - 1
proof
let F be 0-characteristic Field;
let p be non zero Element of the carrier of Polynom-Ring F;
H: Char F = 0 by RING_3:def 6;
per cases;
suppose p is constant; then
  deg p = 0 & (Deriv F).p = 0_.(F) by der4,RING_4:def 4;
  hence thesis by HURWITZ:20;
  end;
suppose p is non constant; then
  A: deg p > 0 by RING_4:def 4; then
  reconsider n = deg p - 1 as Element of NAT by INT_1:3;
  set q = (Deriv F).p;
  B: p.(deg p) = LC p by FIELD_6:2;
  q.n = (n + 1) * p.(n+1) by RINGDER1:def 8
     .= (deg p) '*' p.(deg p) by RING_3:def 2; then
  q.n <> 0.F by H,A,B,REALALG2:25; then
  D: len q >= n + 1 by NAT_1:13,ALGSEQ_1:8;
  deg q < deg p by mm6a; then
  len q - 1 < deg p by HURWITZ:def 2; then
  (len q - 1) + 1 < deg p + 1 by XREAL_1:6; then
  len q <= n + 1 by NAT_1:13; then
  len q = n + 1 by D,XXREAL_0:1;
  hence thesis by HURWITZ:def 2;
  end;
end;
