Consider the Floer cup product with mod 2 coefficients

$$HF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes HF^{0}(\bar{L},\bar{L}^{\prime\prime})\to HF^{0}(\bar{L},\bar{L}),$$

which can be identified as the cup product

$$H^{0}(\bar{L},\bar{L})\otimes H^{0}(\bar{L},\bar{L})\to H^{0}(\bar{L},\bar{L}),\quad 1_{\bar{L}}\cup 1_{\bar{L}}=1_{\bar{L}},$$

and thus must be nontrivial. However, at chain level this Floer product comes from the $A_{\infty}$ operation

$$m_{2}:CF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes CF^{0}(\bar{L},\bar{L}^{\prime\prime})\to CF^{0}(\bar{L},\bar{L}),$$

which must be nontrivial. The counting interpretation implies there are intersection points $p^{\prime}\in CF^{0}(\bar{L},\bar{L}^{\prime\prime})$ and $q^{\prime}\in CF^{0}(\bar{L}^{\prime\prime},\bar{L})\simeq CF^{n}(\bar{L},\bar{L}^{\prime\prime})^{\vee}$ and some holomorphic strip in between. Since degree $0,n$ intersection points cannot occur inside $\mathbb{C}^{n}$, they can only occur at infinity, so we must have $\{p,q\}=\{p^{\prime},q^{\prime}\}$. ∎