
theorem T2:
for L being add-associative right_zeroed right_complementable distributive
            non empty doubleLoopStr
for p,q be Polynomial of L
holds p divides q iff ex r being Polynomial of L st p *' r = q
proof
let L be add-associative right_zeroed right_complementable distributive
         non empty doubleLoopStr;
let p,q be Polynomial of L;
X: now assume p divides q;
   then consider a,b being Element of Polynom-Ring L such that
   B: a = p & b = q & a divides b;
   consider u being Element of Polynom-Ring L such that
   C: a * u = b by B;
   reconsider z = u as Polynomial of L by POLYNOM3:def 10;
   q = p *' z by B,C,POLYNOM3:def 10;
   hence ex r being Polynomial of L st p *' r = q;
   end;
now assume ex r being Polynomial of L st p *' r = q;
  then consider r being Polynomial of L such that
  H: q = p *' r;
  reconsider a=p,b=q,u=r as Element of Polynom-Ring L by POLYNOM3:def 10;
  b = a * u by H,POLYNOM3:def 10;
  then a divides b;
  hence p divides q;
  end;
hence thesis by X;
end;
