
theorem Th30:
  for L being well-unital add-associative right_zeroed
  right_complementable right-distributive non degenerated non empty
  doubleLoopStr, n being Element of NAT holds not(0.L
  is_primitive_root_of_degree n)
proof
  let L be well-unital add-associative right_zeroed right_complementable
  right-distributive non degenerated non empty doubleLoopStr, n be Element of
  NAT;
  defpred P[Nat] means (0.L) |^$1 = 0.L;
A1: for j being Nat st 1 <= j holds P[j] implies P[j+1]
  proof
    let j be Nat;
    assume 1 <= j;
    assume P[j];
    (0.L) |^(j+1) = ((0.L) |^j) * 0.L by GROUP_1:def 7
      .= 0.L;
    hence thesis;
  end;
A2: P[1] by BINOM:8;
A3: for m being Nat st 1 <= m holds P[m] from NAT_1:sch 8(A2,A1);
  assume
A4: 0.L is_primitive_root_of_degree n;
  then n <> 0;
  then 0 + 1 < n + 1 by XREAL_1:8;
  then 1 <= n by NAT_1:13;
  then (0.L) |^n <> 1.L by A3;
  hence contradiction by A4;
end;
