
theorem Lm12b:
for F being Field
for a being Element of F
for n being non zero Nat holds Roots((X-a)`^n) = {a}
proof
let F be Field, a be Element of F, n be non zero Nat;
defpred P[Nat] means Roots((X-a)`^($1)) = {a};
    (X-a)`^1 = (X-a) by POLYNOM5:16; then
IA: P[1] by RING_5:18;
IS: now let k be Nat;
    assume k >= 1;
    assume P[k]; then
    H2: Roots((X-a)`^k) = {a} & Roots(X-a) = {a} by RING_5:18;
    (X-a)`^(k+1) = ((X-a)`^k) *' (X-a) by POLYNOM5:19; then
    Roots((X-a)`^(k+1)) = Roots((X-a)`^k) \/ Roots(X-a) by UPROOTS:23;
    hence P[k+1] by H2;
    end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(IA,IS);
n >= 0 + 1 by INT_1:7;
hence thesis by I;
end;
