(Crossposted from MSE due to no replies)

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part:
$$$$

![enter image description here][1]
$$$$
I give here Proposition 1.1 which the proof uses. 
$$$$
![enter image description here][2]

I do not understand how it's used in the proof. I'd appreciate an explanation. Thank you.


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Here,
$$W_2(0,T) = \{u \in \mathcal V : u' \in \mathcal V'\}$$
and
![enter image description here][3]
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(All images cut from Showalter's book *Monotone Operators in Banach Space*)


  [1]: https://i.sstatic.net/BQ58G.png
  [2]: https://i.sstatic.net/gDNle.png
  [3]: https://i.sstatic.net/3rNxb.png