
theorem mulzero1:
for F being Field,
    p being non zero Polynomial of F
for a being Element of F
for n being Element of NAT
holds (X-a)`^n divides p iff multiplicity(p,a) >= n
proof
let F be Field, p be non zero Polynomial of F;
let a be Element of F, n be Element of NAT;
set m = multiplicity(p,a);
H: (X-a)`^m divides p & not (X-a)`^(m+1) divides p by FIELD_14:67;
now assume B: (X-a)`^n divides p;
   now assume m < n;
     then m + 1 <= n by INT_1:7;
     hence contradiction by H,B,FIELD_14:40;
     end;
   hence multiplicity(p,a) >= n;
   end;
hence thesis by H,FIELD_14:40;
end;
