theorem Th3:
  for T being TopGroup, t being Point of T, f being Loop of t holds
  inverse_loop(f).x * f.x = 1_T
  proof
    let T be TopGroup, t be Point of T, f be Loop of t;
    inverse_loop(f).x = (f.x)" by Th2;
    hence thesis by GROUP_1:def 5;
  end;
