Rank of the $p$-adic closure of $\varphi(\bar{E}(\mathbb{F}))\cdot\langle h_1,\dots,h_m\rangle$ is at most $\mathbb{Z}_p\text{-rk}\,\bar{E}(\mathbb{F}) + m$

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Rank of the $p$-adic closure of $\varphi(\bar{E}(\mathbb{F}))\cdot\langle h_1,\dots,h_m\rangle$ is at most $\mathbb{Z}_p\text{-rk}\,\bar{E}(\mathbb{F}) + m$ · Prove2Me