Prove: If e is an edge in a simple closed path in G, then e belongs to some cycle.

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Cycles in graphs are of two types :

Thus, a simple closed path is always a cycle (but vice versa is not true).

Now, if e is an edge in a simple closed path, means e belongs to a simple closed path in the graph.

And as every simple closed path is a cycle.

Thus, e belongs to some cycle in the graph.

Hence proved.