
theorem
  for F be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, x,y,z being Element of F holds
  x*(y-z) = x*y - x*z
proof
  let F be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr, x,y,z be Element of F;
  x*(y-z) = x*y+x*(-z) by Def2
    .= x*y - x*z by Th4;
  hence thesis;
end;
