Recall that for $L$ to be a special Lagrangian, then its phase angle is constant, and its J-volume is

$$\text{Vol}_{J}(L)=|\int_{L}\Omega|<|\int_{L_{1}}\Omega|+|\int_{L_{2}}\Omega|,$$

using the triangle inequality, the homological relation $[L]=[L_{1}+L_{2}]\in H_{n}(X)$ and the assumption that $\hat{\theta}_{1}>\hat{\theta}_{2}$. Thus the conclusion of the theorem is a quantitative obstruction for $L$ to be special Lagrangian. In section 3.6 we will discuss the relation to Joyce’s LMCF program and the Bridgeland stability condition.