of generality.

\[prop:Krawczyk\] Suppose $K$ is a Krawczyk operator as given in Definition \[def:Krawczyk\] and $ T:= x - A^\dagger f(x)$.

(i) If $x \in X$, then $ T(x) \in K(X,\bar{x})$.

(ii) If $ \hat{x} \in X$ and $ f(\hat{x} )=0$, then $ \hat{x} \in K(X,\bar{x})$.

(iii) If $K(X,\bar{x}) \subseteq X$ and $X$ is compact, then there exists a point $ \hat{x} \in X$ such that $ f(\hat{x})=0$.

(iv) If $K(X,\bar{x}) \subseteq X$ and there exists $0 \leq \lambda < 1$ such that $( I - A^\dagger Df(X))(X-\bar{x}) \subseteq \lambda \cdot ( X - \bar{x})$, then there exists a unique point $ \hat{x} \in X$ such that $ f(\hat{x})=0$.

$\,$

(i) Fix a point $x \in X$ and write $ h = x - \bar{x}$. By the mean-value theorem for Frechet differentiable functions [@ambrosetti1995primer], we have: $$\begin{aligned}
            T(x) 
            &= \bar{x} - A^\dagger f(\bar{x}) + \int_{0}^{1} DT( \bar{x} + t h)  \cdot h \, dt  \nonumber \\ 
            &=  \bar{x} - A^\dagger f(\bar{x}) + \lim_{N \to \infty}    \sum_{i=1}^N  \tfrac{1}{N} \left( I - A^\dagger Df(\bar{x} + \tfrac{i}{N}h) \right) \cdot h  \nonumber \\
            &\in   \bar{x} - A^\dagger f(\bar{x}) + \overline{conv} \left(  \left( I - A^\dagger Df(X) \right) \cdot (x - \bar{x}) \right)   \\
            &\subseteq 