
theorem mm6a:
for R being non degenerated comRing
for p being non zero Element of the carrier of Polynom-Ring R
holds deg (Deriv R).p < deg p
proof
let F be non degenerated comRing,
    p be non zero Element of the carrier of Polynom-Ring F;
set q = (Deriv F).p;
p <> 0_.(F); then
len p <> 0 by POLYNOM4:5; then
reconsider lp = len p - 1 as Element of NAT by INT_1:3;
now let i be Nat;
   assume i >= lp; then
   i + 1 >= (len p - 1) + 1 by XREAL_1:7; then
   p.(i+1) = 0.F by ALGSEQ_1:8;
   hence q.i = (i+1) * 0.F by RINGDER1:def 8 .= 0.F;
   end; then
lp is_at_least_length_of q by ALGSEQ_1:def 2; then
len q <= lp by ALGSEQ_1:def 3; then
len q + 1 <= (len p - 1) + 1 by XREAL_1:7; then
len q < len p by NAT_1:13; then
len q - 1 < len p - 1 by XREAL_1:14;
then deg q < len p - 1 by HURWITZ:def 2;
hence deg q < deg p by HURWITZ:def 2;
end;
