= Solution
Specify the trading conventions first. Let the <bank account> have value $R_r>0$, with $R_0=1$ and its next gross return known at time $r$. Write $B_r=1/R_r$ for the <discount factor>. Let $S_r$ be the ex-dividend asset-price vector, and $d_r$ the vector of cash dividends paid per unit at time $r$; set $d_r=0$ for non-dividend-paying assets. Under an <equivalent martingale measure> $Q$, each discounted cumulative asset gain
$$
G_r=B_rS_r+\sum_{j=1}^rB_jd_j
$$
is a <martingale>. Equivalently, $\mathbb E_Q[B_{r+1}(S_{r+1}+d_{r+1})\mid\mathcal F_r]=B_rS_r$. This is the gain-process form of risk-neutral pricing, rather than a martingale assumption on raw ex-dividend prices.
Let $X_r$ be the $\mathcal F_r$-<measurable> holdings after trading and distributions at time $r$. The previous holdings $X_{r-1}$ earn the next price change and dividends. Define the strategy's net cash payment and its residual <portfolio wealth> by
$$
D_r^X=X_{r-1}^\top(S_r+d_r)-X_r^\top S_r,\qquad V_r=X_r^\top S_r.
$$
A negative payment is an external infusion; a <self-financing portfolio> with no consumption has $D_r^X=0$. The discounted budget equation is
$$
B_rV_r-B_{r-1}V_{r-1}+B_rD_r^X=X_{r-1}^\top(G_r-G_{r-1}).
$$
Assume these gains and payments are <integrable> under $Q$, for example by bounded predictable holdings and integrable discounted asset gains. Then
$$
M_r=B_rV_r+\sum_{j=1}^rB_jD_j^X
$$
is a true <martingale>, by conditioning the displayed one-step gains. Merely assuming a local martingale without appropriate <integrability> would not justify the conditional equality.
Conditioning $M_n$ on $\mathcal F_r$ proves the general <multi-period dividend pricing identity>
$$
\boxed{B_rX_r^\top S_r=\mathbb E_Q\!\left[B_nV_n+\sum_{j=r+1}^nB_jD_j^X\,\middle|\,\mathcal F_r\right].}
$$
For a strategy liquidated at time $n$, $X_n=0$ and all liquidation proceeds are included in $D_n^X$. The terminal value disappears, giving exactly the requested dividend-only relation. That liquidation convention is necessary: holding one unit of the bank account with no payments would otherwise have a positive left side and zero right side.
An <attainable claim> $H$ paid at $n$ can be replicated by a strategy with no earlier payments and final liquidation payment $D_n^X=H$. Its time-$r$ value is therefore
$$
\boxed{V_r=\frac1{B_r}\mathbb E_Q[B_nH\mid\mathcal F_r].}
$$
Two admissible replicating strategies have the same value by this identity, and the value is the same under every <equivalent martingale measure> for which the replication gains are true <martingales>. For unattainable claims this expectation is a possible model price, but the replication argument does not establish a unique price. With density process $Z_r=\mathbb E_P[dQ/dP\mid\mathcal F_r]$, the equivalent <state-price density> form is $V_r=(B_rZ_r)^{-1}\mathbb E_P[B_nZ_nH\mid\mathcal F_r]$.
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