 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 :: TH90
  for G being Group
  for g being Element of G
  for i,j being Integer
  st g |^ i = g |^ j
  holds g |^ (- i) = g |^ (- j)
proof
  let G be Group;
  let g be Element of G;
  let i,j be Integer;
  assume A1: g |^ i = g |^ j;
  thus g |^ (- j) = (g |^ (- j)) * (1_G) by GROUP_1:def 4
                 .= (g |^ (- j)) * (g |^ 0) by GROUP_1:25
                 .= (g |^ (- j)) * (g |^ (i + (- i)))
                 .= (g |^ (- j)) * ((g |^ i) * (g |^ (- i))) by GROUP_1:33
                 .= ((g |^ (- j)) * (g |^ i)) * (g |^ (- i)) by GROUP_1:def 3
                 .= ((g |^ (- j)) * (g |^ j)) * (g |^ (- i)) by A1
                 .= (g |^ ((- j) + j)) * (g |^ (- i)) by GROUP_1:33
                 .= (1_G) * (g |^ (- i)) by GROUP_1:25
                 .= g |^ (- i) by GROUP_1:def 4;
end;
