 Solution

This is the forward implication of the atomic membership truth lemma for forcing. Suppose ${\sigma }^G\in {\tau }^G$. By the evaluation of a forcing name, choose $(\rho ,r)\in \tau$ with $r\in G$ and ${\sigma }^G={\rho }^G$. The assumed equality truth lemma supplies $s\in G$ with $s{⊩}^{\ast }\sigma =\rho$.

Directedness of the generic filter gives $p\in G$ with $p\le r,s$. For every $q\le p$, monotonicity of the syntactic forcing relation gives $q{⊩}^{\ast }\sigma =\rho$, and $q\le r$. Thus $q$ itself witnesses the required membership density below $p$. HenceOnly the equality truth lemma stipulated in the source and the recursive membership clause have been used.