
theorem DP0:
for p being Prime
for R being p-characteristic commutative Ring
for f being Element of the carrier of Polynom-Ring R
holds (Deriv R).f = 0_.(R) iff
      for i being Nat st i in Support f holds p divides i
proof
let p be Prime, R be p-characteristic commutative Ring;
let f be Element of the carrier of Polynom-Ring R;
set q = (Deriv R).f;
A: now assume
   AS: for i being Nat st i in Support f holds p divides i;
   now let j be Nat;
     B: q.j = (j+1) * f.(j+1) by RINGDER1:def 8;
     per cases;
     suppose j + 1 in Support f;
       hence q.j = 0.R by B,AS,Lm2
                .= (0_.(R)).j by ORDINAL1:def 12,FUNCOP_1:7;
       end;
     suppose not j + 1 in Support f;
       hence q.j = (j + 1) * 0.R by B,POLYNOM1:def 4
                .= (0_.(R)).j by ORDINAL1:def 12,FUNCOP_1:7;
       end;
     end;
   hence q = 0_.(R);
   end;
now assume AS: q = 0_.(R);
  now assume ex j being Nat st j in Support f & not p divides j; then
    consider j being Nat such that B: j in Support f & not p divides j;
    C: f.j is non zero by B,POLYNOM1:def 3;
    p * 0 = 0;
    then j <> 0 by B,NAT_D:def 3;
    then reconsider j1 = j - 1 as Nat;
    j1 + 1 = j; then
    q.j1 = j * f.j by RINGDER1:def 8; then
    q.j1 <> 0.R by B,C,Lm2a;
    hence contradiction by AS,ORDINAL1:def 12,FUNCOP_1:7;
    end;
  hence for i being Nat st i in Support f holds p divides i;
  end;
hence thesis by A;
end;
