
theorem
for L being Abelian add-associative right_zeroed right_complementable
            well-unital associative commutative distributive
            almost_left_invertible non trivial doubleLoopStr
for p1,p2 being Polynomial of L
st p1 divides p2 & p1 is with_roots holds p1,p2 have_common_roots
proof
let L be Abelian add-associative right_zeroed right_complementable
  well-unital associative commutative distributive almost_left_invertible
 non trivial doubleLoopStr;
let p1,p2 be Polynomial of L;
assume A1: p1 divides p2 & p1 is with_roots;
per cases;
suppose A2: p1 = 0_.(L);
  p2 mod p1 = 0_.(L) by A1;
  then 0_.(L) = p2 - 0_.(L)by A2,POLYNOM3:34
             .= p2 + 0_.(L) by HURWITZ:9;
  then A3: p2 = 0_.(L) by POLYNOM3:28;
  eval(0_.(L),0.L) = 0.L by POLYNOM4:17;
  then 0.L is_a_root_of 0_.(L) by POLYNOM5:def 7;
  then 0.L is_a_common_root_of p1,p2 by A2,A3;
  hence thesis;
  end;
suppose p1 <> 0_.(L);
  then consider s being Polynomial of L such that
  A4: s *' p1 = p2 by A1,HURWITZ:34;
  consider x being Element of L such that
  A5: x is_a_root_of p1 by A1,POLYNOM5:def 8;
  A6: eval(p1,x) = 0.L by A5,POLYNOM5:def 7;
  eval(p2,x) = eval(s,x) * eval(p1,x) by A4,POLYNOM4:24
            .= 0.L by A6;
  then x is_a_root_of p2 by POLYNOM5:def 7;
  then x is_a_common_root_of p1,p2 by A5;
  hence thesis;
  end;
end;
