Let k, be a theory whose language has only = as a predicate letter and no function letters or individual constants. let its proper axioms be (vx1)x1 = x1, (vxl)(vx2)(x2 = x2 * x2 = xı), and (vx1)(vx2)(vx3)(x1 = x2 = (x2 = xz x1 = x; show that k is a theory with equality. (hint: it suffices to prove that + xy = x3 = (x2 = x2 = x3 = x,) and x2 = x3 = (x1 = x2 = x2 = x3).] k, is called the pure first-order theory of equality.