 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 Th86:
  for n being non zero Nat
  for x being Element of INT.Group n
  holds x" = x |^ (n - 1)
proof
  let n be non zero Nat;
  let x be Element of INT.Group n;
  set N = card (INT.Group n);
  per cases by NAT_1:53;
  suppose n = 1;
    then A1: x = 1_(INT.Group n) by Th77;
    hence x" = (1_(INT.Group n)) by GROUP_1:8
            .= (1_(INT.Group n)) |^ (n - 1) by GROUP_1:31
            .= x |^ (n - 1) by A1;
  end;
  suppose n > 1;
    then A2: 1 mod N = 1 by PEPIN:5;
    thus x" = (x |^ 1)" by GROUP_1:26
           .= x |^ (N - (1 mod N)) by GR_CY_1:10
           .= x |^ (n - 1) by A2;
  end;
end;
