reserve n for Nat;

theorem ro4:
for R being non degenerated Ring,
    a being Element of R holds Roots rpoly(1,a) = {a}
proof
let R be non degenerated Ring, a be Element of R;
set p = rpoly(1,a);
A: now let u be object;
   assume u in {a};
   then A: u = a by TARSKI:def 1;
   eval(p,a) = a - a by HURWITZ:29 .= 0.R by RLVECT_1:15;
   then a is_a_root_of p by POLYNOM5:def 7;
   hence u in Roots(p) by A,POLYNOM5:def 10;
   end;
now let u be object;
  assume B: u in Roots(p);
  then reconsider x = u as Element of R;
  x is_a_root_of p by B,POLYNOM5:def 10;
  then 0.R = eval(p,x) by POLYNOM5:def 7 .= x - a by HURWITZ:29;
  then a = x by RLVECT_1:21;
  hence u in {a} by TARSKI:def 1;
  end;
hence thesis by A,TARSKI:2;
end;
