On simple extensions generated by key polynomials

Let $K^h$ be a Henselization of a valued field $(K,v),$ $w$ a valuation-transcendental extension of $v$ to $K[x],$ and $ϕ$ a key polynomial for $w$. In this paper, we prove that if $ϕ$ is irreducible over $K^h,$ then the simple extension $L|K,$ generated by a root of any key polynomial for $w,$ is u