reserve n for Nat;

theorem Th9:
for R being domRing
for p being Polynomial of R
for x being Element of R holds eval(p,x) = 0.R iff rpoly(1,x) divides p
proof
let L be domRing; let p be Polynomial of L; let x be Element of L;
A1: now assume rpoly(1,x) divides p;
    then consider u being Polynomial of L such that
    A2: rpoly(1,x) *' u = p by RING_4:1;
    A3: eval(rpoly(1,x),x) = x - x by HURWITZ:29 .= 0.L by RLVECT_1:15;
    thus eval(p,x) = eval(rpoly(1,x),x) * eval(u,x) by A2,POLYNOM4:24
                  .= 0.L by A3;
    end;
now assume eval(p,x) = 0.L;
  then consider s being Polynomial of L
  such that A4: p = rpoly(1,x) *' s by HURWITZ:33,POLYNOM5:def 7;
  thus rpoly(1,x) divides p by RING_4:1,A4;
  end;
hence thesis by A1;
end;
