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The rank assumption says that ${\Gamma }^{\prime }=\Gamma \cap V^{\prime }$ is a full lattice in the real vector space underlying $V^{\prime }$. Thus $X^{\prime }=V^{\prime }/{\Gamma }^{\prime }$ is compact, and the inclusion $V^{\prime }\hookrightarrow V$ descends to an injective holomorphic homomorphism $X^{\prime }\hookrightarrow X$. It is therefore a complex subtorus.

Conversely, let $j:Y\hookrightarrow X$ be a subtorus. Its differential at the identity identifies the universal cover of $Y$ with a complex subspace $V^{\prime }\subseteq V$. Lifting $j$ to universal covers shows that the period lattice of $Y$ is precisely $V^{\prime }\cap \Gamma$. Compactness of $Y$ makes this a full lattice of rank $2{\dim }_{\mathbb{C}}{V}^{\prime }$, so every subtorus has the stated form.

Now let $H$ be a polarisation on $X$. Its restriction to $V^{\prime }$ is still positive definite, and its imaginary part remains integral on ${\Gamma }^{\prime }$, so it polarises $X^{\prime }$. Define the Hermitian orthogonal complementBecause $V^{\prime }$ is spanned over $\mathbb{R}$ by lattice vectors and $E=\mathrm{Im}H$ is integral on $\Gamma$, the real equations $E(v,{\gamma }^{\prime })=0$ for ${\gamma }^{\prime }\in {\Gamma }^{\prime }$ make $V^{\prime \prime }$ rational with respect to $\Gamma$. Hence ${\Gamma }^{\prime \prime }=V^{\prime \prime }\cap \Gamma$ is a full lattice in $V^{\prime \prime }$, and $X^{\prime \prime }=V^{\prime \prime }/{\Gamma }^{\prime \prime }$ is a subtorus. Since $V=V^{\prime }\oplus V^{\prime \prime }$, the lattice ${\Gamma }^{\prime }+{\Gamma }^{\prime \prime }$ has finite index in $\Gamma$. Consequently the addition mapis an isogeny of complex tori: it is surjective and has finite kernel. Therefore $X=X^{\prime }+X^{\prime \prime }$ and $X^{\prime }\cap X^{\prime \prime }$ is finite.