 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 ThInverseOpGSquaresToIdentity: :: TH94
  for G being Group
  holds (inverse_op G) * (inverse_op G) = id G
proof
  let G be Group;
  for x being Element of the carrier of G
  holds ((inverse_op G) * (inverse_op G)).x = (id G).x
  proof
    let x be Element of the carrier of G;
    thus ((inverse_op G) * (inverse_op G)).x
     = (inverse_op G).((inverse_op G).x) by FUNCT_2:15
    .= (inverse_op G).(x ") by GROUP_1:def 6
    .= (x ") " by GROUP_1:def 6
    .= (id G).x;
  end;

  hence (inverse_op G) * (inverse_op G) = id G by FUNCT_2:def 8;
end;
