
theorem Th10:
for L being Abelian add-associative right_zeroed right_complementable
            well-unital associative commutative distributive
            almost_left_invertible domRing-like
            non degenerated doubleLoopStr
for p,q being Polynomial of L
for x being Element of L st rpoly(1,x) divides (p*'q)
holds rpoly(1,x) divides p or rpoly(1,x) divides q
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital associative commutative distributive domRing-like
         almost_left_invertible non degenerated doubleLoopStr;
let p,q be Polynomial of L;
let x be Element of L;
assume rpoly(1,x) divides (p*'q);
then eval(p*'q,x) = 0.L by Th9;
then A1: eval(p,x) * eval(q,x) = 0.L by POLYNOM4:24;
per cases by A1,VECTSP_2:def 1;
suppose eval(p,x) = 0.L;
  hence thesis by Th9;
  end;
suppose eval(q,x) = 0.L;
  hence thesis by Th9;
  end;
end;
