# Sub-bipartite Graph perfect matching implies Graph perfect matching?

Knowing that $$G$$ is a Bipartite Graph $$g=(S,T,E)$$ and $$G − {x, y}$$ has a perfect matching* $$\forall x\in S, \forall y\in T$$, how can I demonstrate that a perfect matching for $$G$$ exist?

*A perfect matching is a matching which saturates all the vertices in $$G$$.

Let $$x \in S$$ and $$y \in T$$, then by the Hypothesis there is a perfect matching $$M$$ on $$G-x,y$$. Let $$x*\in S-x$$ and $$y*\in T-y$$ be two vertices connected by an edge in $$M$$. Then by the Hypothesis, there is a perfect Matching $$M*$$on $$G-x*,y*$$. The edge set formed by taking $$M*$$ and adding the edge connecting $$x*$$ to $$y*$$ in $$M$$ is a perfect matching for G.
Take any edge $$xy$$ in $$G$$.
By supposition $$G-\{x,y\}$$ has a perfect matching $$M$$.
Therefore, $$M \cup \{xy\}$$ is a perfect matching for $$G$$.