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Proof step 7 from proof meta-path (previous step)

⊦ ispath(p, n, dirs, files) → meta(p, n, dirs, files) = (lookup(p, n, dirs, files) ∈ dirs ⊃ dirs[lookup(p, n, dirs, files)].meta;files[lookup(p, n, dirs, files)].meta)

Lemma meta