theorem Th1:
  for F being add-associative right_zeroed right_complementable
      distributive unital non empty doubleLoopStr
  for a being Element of F holds
  a|^2 = (-a)|^2
  proof
   let F be add-associative right_zeroed right_complementable
      distributive unital non empty doubleLoopStr;
   let a be Element of F;
    set a2 = -a;
    thus a|^2 = a*a by GROUP_1:51
    .= a2*a2 by VECTSP_1:10
    .= a2|^2 by GROUP_1:51;
  end;
