
theorem Th31:
  for L being add-associative right_zeroed right_complementable
  associative commutative well-unital distributive almost_left_invertible non
degenerated non empty doubleLoopStr, m being Element of NAT, x being Element
of L st x is_primitive_root_of_degree m holds x" is_primitive_root_of_degree m
proof
  let L be add-associative right_zeroed right_complementable associative
  commutative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr;
  let m be Element of NAT;
  let x be Element of L;
  assume
A1: x is_primitive_root_of_degree m;
  then
A2: x <> 0.L by Th30;
A3: for i being Element of NAT st 0 < i & i < m holds x" |^i <> 1.L
  proof
    let i be Element of NAT;
    assume 0 < i & i < m;
    then x |^ i <> 1.L by A1;
    hence thesis by A2,Lm11;
  end;
  x |^ m = 1.L by A1;
  then
A4: x" |^ m = 1.L by A2,Lm10;
  m <> 0 by A1;
  hence thesis by A4,A3;
end;
