
theorem
for p being Prime
for R being p-characteristic commutative Ring
for n being non zero Nat
holds { m * 1.R where m is Nat : m < p } c= Roots X^(p|^n,R)
proof
let p be Prime, R be p-characteristic commutative Ring;
let n be non zero Nat;
defpred P[Nat] means ($1) * 1.R in Roots X^(p|^n,R);
    0 * 1.R = (0.R)|^(p|^n) by BINOM:12
           .= (0 * 1.R)|^(p|^n) by BINOM:12; then
    (0 * 1.R) is_a_root_of X^(p|^n,R) by thXX; then
IA: P[0] by POLYNOM5:def 10;
reconsider p1 = p - 1 as Element of NAT by INT_1:3;
IS: now let k be Element of NAT;
    assume 0 <= k & k < p1;
    assume P[k]; then
    (k * 1.R) is_a_root_of X^(p|^n,R) by POLYNOM5:def 10; then
    IV: (k * 1.R)|^(p|^n) = k * 1.R by thXX;
    ((k+1) * 1.R)|^(p|^n)
        = (k * 1.R + 1 * 1.R)|^(p|^n) by BINOM:15
       .= (k * 1.R + 1.R)|^(p|^n) by BINOM:13
       .= (k * 1.R)|^(p|^n) + (1.R)|^(p|^n) by FIELD_15:41
       .= (k * 1.R) + (1 * 1.R) by IV,BINOM:13
       .= (k+1) * 1.R by BINOM:15; then
    ((k+1) * 1.R) is_a_root_of X^(p|^n,R) by thXX;
    hence P[k+1] by POLYNOM5:def 10;
    end;
I: for k being Element of NAT st 0 <= k & k <= p1 holds P[k]
   from INT_1:sch 7(IA,IS);
now let o be object;
  assume o in { i * 1.R where i is Nat : i < p }; then
  consider i being Nat such that
  A: o = i * 1.R & i < p;
  reconsider i as Element of NAT by ORDINAL1:def 12;
  i + 1 <= p by A,INT_1:7;
  then i + 1 - 1 <= p - 1 by XREAL_1:9;
  hence o in Roots X^(p|^n,R) by I,A;
  end;
hence thesis;
end;
