
theorem split1:
for F being Field
for n being non zero Nat
for a,b being Element of F holds b is_a_root_of X^(n,a) iff b|^n = a
proof
let F be Field, n be non zero Nat, a,b be Element of F;
A: now assume b is_a_root_of X^(n,a);
   then 0.F = b|^n - a by teval;
   hence b|^n = a by RLVECT_1:21;
   end;
now assume b|^n = a;
  then 0.F = b|^n - a by RLVECT_1:15 .= eval(X^(n,a),b) by teval;
  hence b is_a_root_of X^(n,a);
  end;
hence thesis by A;
end;
