Theorem: Let T be the paring produced by TMA. T is stable.
- Let b and g be any couple in T.
- Suppose b prefers g* to g. We will argue that g* prefers her husband to b.
- During TMA, b proposed to g* before he proposed to g. Hence, at some point g* rejected b for someone she preferred. By the MAYBE lemma, the person she married was also preferable to b.
- Thus, every boy will be rejected by any girl he prefers to his wife. T is stable.