) if and only if $V$ is defined as in for some clearing payments $p \in [0,\bar p]$ as defined in the fixed point problem .

We will prove the first equivalence only, the second follows similarly.

Let $p \in [0,\bar p]$ be a clearing payment vector. Define the wealth vector $V$ by , then it is clear that $V^- = \bar p - p$ by definition as well, i.e., $p = \bar p - V^- \geq 0$. Thus from we immediately recover that the wealth vector $V$ must satisfy .

Let $p = [\bar p - V^-]^+$ for some wealth vector $V \in {\mathbb{R}}^{n+1}$ satisfying . By construction we find $$\begin{aligned}
p &= [\bar p - V^-]^+ = \bar p - \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-
= \bar p - \left(x + \Pi^{\top}p - \bar p\right)^- = \bar p \wedge \left(x + \Pi^{\top}p\right).\end{aligned}$$ We note that $\bar p \geq \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-$ can be shown trivially.

Due to the equivalence of the clearing payments and clearing wealths provided in Proposition \[prop:EN-e\], we are able to consider the Eisenberg-Noe system as a fixed point of equity and losses rather than 