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What if you write $$P[R_{n+1} = d|F_n] = 1 - P[R_{n+1} = u|F_n] ?$$ Let us write $P(u) = P[R_{n+1} = u|F_n]$ Then the part to show is $$u \bar{S}_n P(u) + d \bar{S}_n (1-P(u))$$ and this $$\bar{S}_n \left(d +(u-d)P(u) \right),$$ where we just expanded terms and then extracted the coefficients.