
theorem Th9:
for L being Abelian add-associative right_zeroed right_complementable
            well-unital associative commutative distributive
            almost_left_invertible non degenerated doubleLoopStr
for p being Polynomial of L
for x being Element of L
holds eval(p,x) = 0.L iff rpoly(1,x) divides p
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital associative commutative distributive
         almost_left_invertible non degenerated doubleLoopStr;
let p be Polynomial of L;
let x be Element of L;
A1: now assume rpoly(1,x) divides p;
   then p mod rpoly(1,x) = 0_.(L);
   then (p - (p div rpoly(1,x)) *' rpoly(1,x))
                               + (p div rpoly(1,x)) *' rpoly(1,x)
         = (p div rpoly(1,x)) *' rpoly(1,x) by POLYNOM3:28;
   then A2: (p div rpoly(1,x)) *' rpoly(1,x)
     = p + (-(p div rpoly(1,x)) *' rpoly(1,x)
                 + (p div rpoly(1,x)) *' rpoly(1,x)) by POLYNOM3:26
    .= p + ( ((p div rpoly(1,x)) *' rpoly(1,x))
           - ((p div rpoly(1,x)) *' rpoly(1,x)) )
    .= p + 0_.(L) by POLYNOM3:29
    .= p by POLYNOM3:28;
   A3: eval(rpoly(1,x),x) = x - x by HURWITZ:29 .= 0.L by RLVECT_1:15;
   thus eval(p,x) = eval(p div rpoly(1,x),x) * 0.L by A3,A2,POLYNOM4:24
                 .= 0.L;
   end;
   eval(p,x) = 0.L implies rpoly(1,x) divides p by HURWITZ:35,POLYNOM5:def 7;
 hence thesis by A1;
end;
