
theorem Th32:
  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 for i,j being Nat st 1 <= i & i <= m &
  1 <= j & j <= m & i <> j holds pow(x,i-j) <> 1.L
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;
  let i,j be Nat;
  assume that
A3: 1 <= i and
A4: i <= m and
A5: 1 <= j and
A6: j <= m and
A7: i <> j;
  per cases;
  suppose
A8: i > j;
    then reconsider k = i - j as Element of NAT by INT_1:5;
    k <= i - 1 by A5,XREAL_1:13;
    then k + 1 <= (i - 1) + 1 by XREAL_1:6;
    then k < i by NAT_1:13;
    then
A9: k < m by A4,XXREAL_0:2;
    i - j > j - j by A8,XREAL_1:14;
    then x|^k <> 1.L by A1,A9;
    hence thesis by Def2;
  end;
  suppose
    i <= j;
    then
A10: i < j by A7,XXREAL_0:1;
    then
A11: j - i > i - i by XREAL_1:14;
A12: i - j < j - j by A10,XREAL_1:14;
    then
A13: |.i-j.| = -(i-j) by ABSVALUE:def 1;
    then reconsider k = -(i-j) as Element of NAT;
    j - i <= j - 1 by A3,XREAL_1:13;
    then k + 1 <= (j - 1) + 1 by XREAL_1:6;
    then k < j by NAT_1:13;
    then
A14: k < m by A6,XXREAL_0:2;
A15: x|^k <> 0.L by A2,Th1;
    now
      assume (x|^k)" = 1.L;
      then 1.L = x|^k * 1.L by A15,VECTSP_1:def 10
        .= x|^k;
      hence contradiction by A1,A11,A14;
    end;
    hence thesis by A12,A13,Def2;
  end;
end;
