Suppose that in the definition of a group G$,$ the condition that for each element a in G there exists an element a$^-1$ in G with the property $($a$)($a$^-1)(=$a$^-1)($a$)=$e is replaced by the condition $($a$)($a$^-1)=$e$.$ Show that $($a$^-1)($a$)=$e$.$