 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem ThINTGroupOrd3: :: TH92
  for n being non zero Nat
  for g1 being Element of INT.Group n st g1 = 1
  for i,j being Nat
  holds g1 |^ i = g1 |^ j iff i mod n = j mod n
proof
  let n be non zero Nat;
  let g1 be Element of INT.Group n;
  assume A1: g1 = 1;
  let i,j be Nat;
  thus g1 |^ i = g1 |^ j implies i mod n = j mod n
  proof
    assume A2: g1 |^ i = g1 |^ j;
    thus i mod n = g1 |^ i by A1,LmINTGroupOrd3
                .= g1 |^ j by A2
                .= j mod n by A1,LmINTGroupOrd3;
  end;
  assume A3: i mod n = j mod n;
  thus g1 |^ i = g1 |^ (n * (i div n) + (i mod n)) by NAT_D:2
       .= (g1 |^ (n * (i div n))) * (g1 |^ (i mod n)) by GROUP_1:33
       .= ((g1 |^ (card (INT.Group n))) |^ (i div n)) *
            (g1 |^ (i mod n)) by GROUP_1:35
       .= ((1_(INT.Group n)) |^ (i div n)) * (g1 |^ (i mod n))
            by GR_CY_1:9
       .= (1_(INT.Group n)) * (g1 |^ (i mod n)) by GROUP_1:31
       .= ((1_(INT.Group n)) |^ (j div n)) * (g1 |^ (i mod n))
            by GROUP_1:31
       .= ((1_(INT.Group n)) |^ (j div n)) * (g1 |^ (j mod n)) by A3
       .= ((g1 |^ (card (INT.Group n))) |^ (j div n)) *
            (g1 |^ (j mod n)) by GR_CY_1:9
       .= (g1 |^ (n * (j div n))) * (g1 |^ (j mod n)) by GROUP_1:35
       .= g1 |^ (n * (j div n) + (j mod n)) by GROUP_1:33
       .= g1 |^ j by NAT_D:2;
end;
