Theorem: The TMA pairing, T, is female-pessimal.
We know it is male-optimal. Suppose there is a stable pairing S where some girl g does worse than in T.
Let b be her mate in T. Let b* be her mate in S.
- By assumption, g likes b better than her mate in S
- b likes g better than his mate in S
- We already know that g is his optimal girl
- Therefore, S is not stable.