Let $V, V'$ be full flags in $\mathbb{C}^n$, let $\lambda, \mu$ be admissible partitions, and let $$\sigma_\lambda(V) = \{\Lambda \in G(k,n): \dim(\Lambda \cap V_{n-k+i-\lambda_i}) \geq i\}$$ and $$\sigma_\mu(V) = \{\Lambda \in G(k,n): \dim(\Lambda \cap V_{n-k+i-\mu_i}) \geq i\}$$ be general Schubert cycles. Then for each $i$ and any $\Lambda \in \sigma_\lambda(V) \cap \sigma_\mu(V')$, we have $$\dim(\Lambda \cap V_{n-k+i-\lambda_i}) \geq i$$ and $$\dim(\Lambda \cap V'_{n-k+(k-i+1)-b_{k-i+1}})\geq k-i+1.$$ This implies that $$\Lambda \cap V_{n-k+i-\lambda_i} \cap V'_{n-i+1-b_{k-i+1}} \neq 0.$$ I don't understand why the last statement is true. I assume we have to show that $$\dim(\Lambda \cap V_{n-k+i-\lambda_i} \cap V'_{n-i+1-b_{k-i+1}}) \geq 1,$$ but how can this be shown?
I would appreciate any help. Thank you!