Stability preservation theorems

The basic result of the paper is the main theorem worded as follows.
Let $\mathbb F=\langle F,R\rangle$ be a valued field such that $\mathbb F_R$ has characteristic $p>0$ and let $\mathbb F_0\ge\mathbb F$ be an extension of valued fields satisfying the following conditions:
(i) there exists a set $B_0\subset R_0\setminus\mathfrak m(R_0)$ for which $\overline B_0\rightleftharpoons\{\overline b\rightleftharpoons b+\mathfrak m(R_0)\mid b\in B_0\}$ is a separating transcendence basis for a field $F_{R_0}$ over $F_R$;
(ii) $\Gamma_R$ is $p$-pure in $\Gamma_{R_0}$, i.e., $\Gamma_{R_0}/\Gamma_R$ does not contain elements of order $p$;
(iii) there exists a set $B_1\subset F^\times_0$ such that the family $\widetilde B_1\rightleftharpoons\{\widetilde b\rightleftharpoons v_{R_0}(b)+(p\Gamma_{R_0})\Gamma_R\mid b\in B_1\}$ is linearly independent in the elementary $p$-group $\Gamma_{R_0}/(p\Gamma_{R_0})\Gamma_R$;
(iv) $F_0$ is algebraic over $F(B_0\cup B_1)$.
Then the property of being stable for $\mathbb F$ implies being stable for $\mathbb F_0$.