
theorem Lm12a:
for F being Field
for a being Element of F
for n being Nat holds deg((X-a)`^n) = n
proof
let F be Field, a be Element of F, n be Nat;
defpred P[Nat] means deg((X-a)`^($1)) = $1;
H: (X-a)`^0 = 1_.(F) by POLYNOM5:15;
IA: P[0] by H,RATFUNC1:def 2;
IS: now let k be Nat;
    assume P[k]; then
    H2: deg((X-a)`^k) = k & deg(X-a) = 1 by FIELD_5:def 1;
    (X-a)`^(k+1) = ((X-a)`^k) *' (X-a) &
       X-a <> 0_.(F) & (X-a)`^k <> 0_.(F) by POLYNOM5:19;
    hence P[k+1] by H2,HURWITZ:23;
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
