Lines in flags

Let F_1 \subseteq \dots \subseteq F_n and F'_1 \subseteq \dots \subseteq F'_n be complete flags over an n-dimensional vector space V. Show that there is a set of n lines L_1,\dots, L_n compatible with the two flags, in the sense that there exists a permutation \sigma \in S_n such that F_i = L_1 \oplus \dots \oplus L_i and F'_i = L_{\sigma(1)}\oplus \dots \oplus L_{\sigma(i)}.

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Lines in flags