Solution
= Solution
Take $(x,y)\in E_r(K)\setminus\{O\}$. Because $v(x)<0$ and $p,q$ have degrees $d,d-1$, their leading terms dominate, giving
$$
v\left(\frac{p(x)}{q(x)}\right)=v(x)\leq-2r.
$$
The polynomial $p'q-pq'$ has degree $2d-2$ with nonzero leading coefficient, the same degree as $q^2$. Their quotient consequently has valuation zero at $x$, so
$$
v\left(\frac{p'q-pq'}{cq^2}y\right)=v(y)\leq-3r.
$$
The isogeny sends $O$ to $O$, hence $\phi(E_r(K))\subseteq E'_r(K)$.