
theorem ZZ1a:
for F being Field
for S being non empty finite Subset of F
for p being Ppoly of F,S
for q being non zero with_roots Polynomial of F
st p *' q is Ppoly of F,(S \/ Roots q) holds q is Ppoly of F,(Roots q)
proof
let F be Field, S be non empty finite Subset of F;
let p be Ppoly of F,S; let q be non zero with_roots Polynomial of F;
assume AS: p *' q is Ppoly of F,(S \/ Roots q); then
   q *' p is Ppoly of F,(S \/ Roots q) & q is monic by ZZ3y; then
A: q is Ppoly of F by FIELD_8:10;
   S = Roots p by RING_5:63; then
H: S \/ Roots q = Roots(p*'q) by UPROOTS:23;
now let a be Element of F;
  assume B: a is_a_root_of q;
  D: multiplicity(q,a) >= 1 by B,UPROOTS:52;
  now assume E: multiplicity(q,a) > 1;
    multiplicity(q,a) <= multiplicity(p*'q,a) by ZZ7;
    hence contradiction by B,H,E,AS,ZZ1,ro1;
    end;
  hence multiplicity(q,a) = 1 by D,XXREAL_0:1;
  end;
hence thesis by A,ZZ1;
end;
