
theorem lemppolspl2:
for F being Field,
    p being Polynomial of F
for a being non zero Element of F
holds a * p splits_in F iff p splits_in F
proof
let F be Field, p be Polynomial of F; let a be non zero Element of F;
X: now assume p splits_in F; then
   consider b being non zero Element of F, q being Ppoly of F such that
   A: p = b * q by FIELD_4:def 5;
   a * p = (a * b) * q by A,RING_4:11;
   hence a * p splits_in F by FIELD_4:def 5;
   end;
now assume a * p splits_in F; then
   consider b being non zero Element of F, q being Ppoly of F such that
   A: a * p = b * q by FIELD_4:def 5;
   a" <> 0.F & b <> 0.F by VECTSP_2:13; then
   B: a" * b is non zero by VECTSP_2:def 1;
   a <> 0.F; then
   a" * a = 1.F by VECTSP_1:def 10; then
   p = (a" * a) * p
    .= a" * (a * p) by RING_4:11
    .= (a" * b) * q by A,RING_4:11;
   hence p splits_in F by B,FIELD_4:def 5;
   end;
hence thesis by X;
end;
