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Let E₀, E₁ be two subsets of the closure D̅ of a domain D of the Euclidean n-space ${ℝ}^n$ and Γ(E₀,E₁,D) the family of arcs joining E₀ to E₁ in D. We establish new cases of equality $M_p\Gamma (E₀,E₁,D)=ca{p}_p(E₀,E₁,D)$, where $M_p\Gamma (E₀,E₁,D)$ is the p-module of the arc family Γ(E₀,E₁,D), while $ca{p}_p(E₀,E₁,D)$ is the p-capacity of E₀,E₁ relative to D and p > 1. One of these cases is when p = n, E̅₀ ∩ E̅₁ = ∅, $E_i=E_i^{\text{'}}\cup E_i^{\text{'}\text{'}}\cup E_i^{\text{'}\text{'}\text{'}}\cup F_i$, $E_i^{\text{'}}$ is inaccessible from D by rectifiable arcs, $E_i^{\text{'}\text{'}}$ is open relative to D̅ or to the boundary ∂D of D, $E_i^{\text{'}\text{'}\text{'}}$ is at most countable, $F_i$ is closed (i = 0,1) and D...