
theorem multi4b:
for F being Field,
    p being non zero Polynomial of F
for q being Polynomial of F
for a being Element of F
st p = ((X-a)`^multiplicity(p,a)) *' q holds eval(q,a) <> 0.F
proof
let F be Field, p be non zero Polynomial of F;
let q be Polynomial of F; let a be Element of F;
assume AS: p = ((X-a)`^multiplicity(p,a)) *' q;
set n = multiplicity(p,a);
now assume eval(q,a) = 0.F;
  then consider r being Polynomial of F such that
  A: q = rpoly(1,a) *' r by HURWITZ:33,POLYNOM5:def 7;
  p = (((X-a)`^n) *' (X-a)) *' r by AS,A,POLYNOM3:33
   .= ((X-a)`^(n+1)) *' r by POLYNOM5:19;
  hence contradiction by FIELD_14:67,RING_4:1;
  end;
hence thesis;
end;
