
theorem ro0:
for F being Field
for a being Element of F, n being Nat holds multiplicity((X-a)`^n,a) = n
proof
let F be Field, a be Element of F, n be Nat;
set p = (X-a)`^n, m = multiplicity(p,a);
I: n + 0 < n + 1 by XREAL_1:6;
(1_.(F)) *' p = p; then
A: (X-a)`^n divides p by RING_4:1;
deg((X-a)`^(n+1)) = n+1 & deg((X-a)`^n) = n by Lm12a;
hence thesis by A,I,RING_5:13,FIELD_14:67;
end;
