
theorem lemppolspl1:
for F being Field,
    p,q being Polynomial of F
st p splits_in F & q splits_in F holds p *' q splits_in F
proof
let F be Field, p,q be Polynomial of F;
assume AS: p splits_in F & q splits_in F; then
consider b being non zero Element of F, u being Ppoly of F such that
A: p = b * u by FIELD_4:def 5;
consider c being non zero Element of F, v being Ppoly of F such that
B: q = c * v by AS,FIELD_4:def 5;
D: u *' v is Ppoly of F by RING_5:52;
p *' q = c * ((b * u) *' v) by A,B,RING_4:10
      .= c * (b * (u *' v)) by RING_4:10
      .= (c * b) * (u*' v) by RING_4:11
      .= (b * c) * (u*' v) by GROUP_1:def 12;
hence thesis by D,FIELD_4:def 5;
end;
