
theorem simpAgcd1:
for F being Field
for S1 being non empty finite Subset of F
for p being Ppoly of F,S1
for q being non constant monic Polynomial of F holds
q divides p iff
ex S2 being non empty finite Subset of F st q is Ppoly of F,S2 & S2 c= S1
proof
let F be Field, S1 be non empty finite Subset of F, p be Ppoly of F,S1;
let q be non constant monic Polynomial of F;
now assume q divides p; then
  consider B2 being non zero bag of the carrier of F such that
  B: q is Ppoly of F,B2 & B2 divides (Bag S1) by ppolydiv;
  consider S2 being non empty finite Subset of F such that
  C: B2 = Bag S2 & S2 c= S1 by B,bagset2;
  thus ex S2 being non empty finite Subset of F
                    st q is Ppoly of F,S2 & S2 c= S1 by B,C;
  end;
hence thesis by ppolydiv,bagset1;
end;
