Let $T_{0},T_{1}$ be [[Trees#^suslin|Suslin trees]], and consider them as the forcing notions [[Forcing trees#Killing an $ omega_{1}$-Souslin tree|adding a branch to themselves]]. Let $\mathcal{S}(T_{i})$ be the [[Specializing an Aronszajn tree with finite conditions|standard specializing poset]] for $T_{i}$. We consider the relationship between these forcing notions. Recall: Lemma A: ![[Chain conditions#^eaac15]] Lemma B: ![[Specializing an Aronszajn tree with finite conditions#^0008e0]] If $T_{1} \otimes T_{2}$ is Suslin then - Lemma A implies $T_{i} \Vdash T_{1-i}$ is Suslin (and in particular Aronszajn). - This and Lemma B imply $T_{i} \Vdash \mathcal{S}(T_{1-i})$ satisfies ccc. - Since the forcings themselves are absolute, this and Lemma A also imply $\mathcal{S}(T_{1-i}) \Vdash T_{i}$ satisfies ccc (i.e. is Suslin). Since the conditions are symmetric, we get: **Lemma.** If $T_{1} \otimes T_{2}$ is Suslin then forcing with $T_{i}$ or $\mathcal{S}(T_{i})$ preserves the Suslinity of $T_{1-i}$ and the ccc of $\mathcal{S}(T_{1-i})$.