Let c ∈ Z: Write a detailed structured proof to prove the statement:

If `c^5 + 7 is even, then c is odd`.

I started out like this:

```
Assume c ∈ Z
    Assume c^5 + 7 == 2n
        Then c == 2n + 1

```

Also, is this claim true? I plugged in odd numbers for c, and haven't encountered a counter-example. Is there a way to determine the veracity of the claim before doing the proof?