I have seen a proof of [Tutte's theorem][1] from Gallai's lemma. Lovasz also said in his [_Matching Theory_][2] that Gallai's lemma can be easily proven from Tutte's theorem. But I cannot figure out how. 

![Matching Theory, by Lovasz and Plummer, p. 89][3]


> **3.1.13. THEOREM.** (_Gallai's Lemma_). _If graph $G$ is connected and $\nu(G-u)=\nu(G)$ for each $u \in V(G)$, then $G$ is factor-critical._  
> We remark that an easy proof would follow from Tutte's Theorem, but here we choose a more direct proof based on Corollary 3.1.7.

Thanks!


  [1]: http://en.wikipedia.org/wiki/Tutte_theorem
  [2]: http://books.google.com/books?id=yW3WSVq8ygcC
  [3]: https://i.sstatic.net/U4HOQ.png