
theorem thXX:
for R being unital non degenerated Ring
for n being non trivial Nat
for a being Element of R holds a is_a_root_of X^(n,R) iff a|^n = a
proof
let R be unital non degenerated Ring,
    n be non trivial Nat, a be Element of R;
A: now assume a is_a_root_of X^(n,R);
   then 0.R = a|^n - a by teval;
   hence a|^n = a by RLVECT_1:21;
   end;
now assume a|^n = a;
  then 0.R = a|^n - a by RLVECT_1:15 .= eval(X^(n,R),a) by teval;
  hence a is_a_root_of X^(n,R);
  end;
hence thesis by A;
end;
