Let $G$ be a graph with at least 1-factor. Prove that if $G$ is 2-edge-connected, then $G$ has at least two perfect matchings.
i tried it by Tutte's theorem first, I choose $e$(edge) in first perfect matching, and when G doesn't have two perfect matching, then there exist $X$ in $G$ ,$odd(G-e-X)>|X|$ by pareity, $odd(G-X)=|X|$
and i was blocked here. how can i solved it?