
theorem thXXe:
for R being unital non degenerated Ring
for S being RingExtension of R
for n being non trivial Nat
for a being Element of S holds a is_a_root_of X^(n,R),S iff a|^n = a
proof
let R be unital non degenerated Ring, S be RingExtension of R,
    n be non trivial Nat, a be Element of S;
A: now assume a is_a_root_of X^(n,R),S;
   then 0.S = Ext_eval(X^(n,R),a) by FIELD_4:def 2
           .= a|^n - a by tevale;
   hence a|^n = a by RLVECT_1:21;
   end;
now assume a|^n = a;
  then 0.S = a|^n - a by RLVECT_1:15
          .= Ext_eval(X^(n,R),a) by tevale;
  hence a is_a_root_of X^(n,R),S by FIELD_4:def 2;
  end;
hence thesis by A;
end;
