Lemma 17 of Lee (2024). Fix $U₀ ∈ 𝒢_l X$ with $g_{T∘P}(U₀) > 0$; then $P$ is injective on $U₀$ and $U := P U₀ ≤ V$ satisfies $g_T(U) ≥ g_{T∘P}(U₀)/‖P‖$. For a unit vector $x ∈ U ⊕ E$, the splitting $x = P x + (1 - P) x$ is carried by $T$ to the splitting along $P'$, so $‖T x‖ ≥ g_T(U) max{‖P x‖, ‖(1 - P) x‖}/(1 + ‖P'‖)$, using $g_T(E) > ‖T|_V‖ ≥ g_T(U)$. Since $max{‖P x‖, ‖(1 - P) x‖} ≥ 1/2$ and $1 + ‖P'‖ ≤ 2‖P'‖$, the $(l + k)$-dimensional space $U ⊕ E$ has $g_T(U ⊕ E) ≥ g_{T∘P}(U₀)/(4‖P‖‖P'‖)$, and taking the supremum over $U₀$ gives the bound.