Don't understand this proof of equivalence of weak solutions to PDE

Question

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: $$$$

$$$$ I give here Proposition 1.1 which the proof uses. $$$$

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

(All images cut from Showalter's book Monotone Operators in Banach Space)