Solution
= Solution
The identity and inverse axioms give
$$
\boxed{\varphi^r(x,0)=x^r,\qquad
\varphi^r(0,x)=x^r,\qquad
\varphi^r(\bar x,x)=0}.
$$
= Solution
The identity and inverse axioms give
$$
\boxed{\varphi^r(x,0)=x^r,\qquad
\varphi^r(0,x)=x^r,\qquad
\varphi^r(\bar x,x)=0}.
$$