Properties of accessible subrings of pseudonormed rings when taking quotient rings

Let $(R,\xi)$ and $(\bar R,\bar\xi)$ be pseudonormed rings, $\varphi\colon R\to\bar R$ be a ring isomorphism. We prove that $\varphi\colon(R,\xi)\to(\bar R,\bar\xi)$ is a superposition of a finite number of semi-isometric isomorphisms if and only if it is a narrowing on an accessible subring of some isometric homomorphism.