
theorem ppolydiv:
for F being Field
for B1 being non zero bag of the carrier of F
for p being Ppoly of F,B1
for q being non constant monic Polynomial of F
holds q divides p iff
      ex B2 being non zero bag of the carrier of F
                                    st q is Ppoly of F,B2 & B2 divides B1
proof
let F be Field, B1 be non zero bag of the carrier of F;
let p be Ppoly of F,B1; let q be non constant monic Polynomial of F;
A: now assume q divides p; then
   consider u being Polynomial of F such that A1: p = q *' u by RING_4:1;
   A2: u is non zero by A1; then
   reconsider q1 = q as Ppoly of F by A1,FIELD_8:10;
   A3: B1 = BRoots p by RING_5:55;
   q1 is Ppoly of F,(BRoots q1) by RING_5:59;
   hence ex B2 being non zero bag of the carrier of F
                   st q is Ppoly of F,B2 & B2 divides B1 by A3,A1,A2,ZZ1fB;
   end;
now let B1 be non zero bag of the carrier of F, p be Ppoly of F,B1;
   let q be non constant monic Polynomial of F;
   assume ex B2 being non zero bag of the carrier of F
                                st q is Ppoly of F,B2 & B2 divides B1; then
   consider B2 being non zero bag of the carrier of F such that
   I3: q is Ppoly of F,B2 & B2 divides B1;
   set B3 = B1 -' B2;
   per cases;
      suppose B3 is zero; then
        p = q *' (1_.(F)) by I3,bag6,bag5;
        hence q divides p by RING_4:1;
        end;
      suppose B3 is non zero; then
        reconsider B3 as non zero bag of the carrier of F;
        set u = the Ppoly of F,B3;
        B3 + B2 = B1 by I3,PRE_POLY:47; then
        reconsider r = q *' u as Ppoly of F,B1 by I3,RING_5:58;
        r = p by bag5;
        hence q divides p by RING_4:1;
        end;
    end;
hence thesis by A;
end;
