ILL 1982 Problem 18

You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that $$(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},$$ where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$ , where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.