reserve i,j,k for Nat;

theorem Th13:
  for K being left_zeroed right_zeroed add-associative
  right_complementable non empty addLoopStr holds comp K is_an_inverseOp_wrt
  the addF of K
proof
  let K be left_zeroed right_zeroed add-associative right_complementable non
  empty addLoopStr;
  now
    let a be Element of K;
    thus (the addF of K).(a,((comp K)).a) = a+ -a by VECTSP_1:def 13
      .= 0.K by RLVECT_1:5
      .= the_unity_wrt the addF of K by Th7;
    thus (the addF of K).(((comp K)).a,a) = -a+a by VECTSP_1:def 13
      .= 0.K by RLVECT_1:5
      .= the_unity_wrt the addF of K by Th7;
  end;
  hence thesis by FINSEQOP:def 1;
end;
