
theorem caBR:
for F being Field
for a being Element of F, n being Nat holds card BRoots((X-a)`^n) = n
proof
let F be Field, a be Element of F, n be Nat;
defpred P[Nat] means card BRoots((X-a)`^($1)) = ($1);
    0 = deg((X-a)`^0) by Lm12a .= len((X-a)`^0) - 1 by HURWITZ:def 2; then
IA: P[0] by UPROOTS:57;
IS: now let k be Nat;
    assume IV: P[k];
    (X-a)`^(k+1) = ((X-a)`^k) *' (X-a) by POLYNOM5:19; then
    card BRoots((X-a)`^(k+1))
       = card BRoots((X-a)`^k) + card BRoots(X-a) by RING_5:41
      .= k + 1 by IV,RING_5:40;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
