(Thomas-Yau existence) Assume the derived Fukaya class of the quantitatively almost calibrated Lagrangian brane $L_{0}$ is Thomas-Yau semistable (cf. Definition 3.32). Then there is a special Lagrangian representative in the same $D^{b}Fuk(X)$ class (or some weaker $S$-equivalence relation, see remark 5.28).