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In the second causal directed acyclic graph, $Y(a_1,a_2)$ depends on $U_2$ but not on $U_1$, whereas $A_1$ depends on $U_1$ and the two latent roots are independent. HenceAlthough conditioning on $X$ conveys information about $U_2$, the assignment $A_2$ uses only $(A_1,X)$ and fresh randomization, soThese are the two sequential exchangeability conditions. Using them successively, together with consistency of potential outcomes, giveswhich is the formula from part a.

Adding $X\to Y$ invalidates the argument in general. The latent variable $U_1$ confounds $A_1$ and $X$, so $\mathbb{P}(X\mid A_1=a_1)$ need not equal the distribution of $X(a_1)$. When $X$ directly affects $Y$, that discrepancy no longer cancels after summing over $x$. The same observed distribution can then correspond to different intervention means, so the displayed formula need not identify the effect.

Writefor the observed history just before $A_t$. A sufficient condition is sequential exchangeabilityfor every treatment regime, together with consistency of potential outcomes and positivity in causal inference. Repeated conditioning then gives the longitudinal G-formulaGraphically, it is enough that each $A_t$ be D-separated from the final counterfactual under the specified regime after conditioning on its observed past. The two independences used in part b are precisely the $T=2$ instance.