
theorem
for p being Prime
for R being p-characteristic commutative Ring
for a being Element of R holds (Deriv R).(X^(p,a)) = 0_.(R)
proof
let p be Prime, R be p-characteristic commutative Ring, a be Element of R;
set q = X^(p,a), D = (Deriv R).(X^(p,a));
now let j be Nat;
  B: D.j = (j+1) * q.(j+1) by RINGDER1:def 8;
  per cases;
  suppose j + 1 in Support q; then
    q.(j+1) <> 0.R by POLYNOM1:def 3; then
    j+1 = p or j+1 = 0 by Lm11;
    hence D.j = 0.R by B,Lm2
             .= (0_.(R)).j by ORDINAL1:def 12,FUNCOP_1:7;
    end;
  suppose not j + 1 in Support q;
    hence D.j = (j + 1) * 0.R by B,POLYNOM1:def 4
             .= (0_.(R)).j by ORDINAL1:def 12,FUNCOP_1:7;
    end;
  end;
 hence D = 0_.(R);
end;
