Cauchy–Binet conjecture for determinant-like maps on skew partial fields

Let $\mathbb{P}=(R,G)$ be a skew partial field, let $n,r,s$ be positive integers with $s\geq r$, define

$$X:=\{1,\ldots,r\},\qquad E:=\{1,\ldots,s\},$$

and let $A$ be an $X\times E$ weak $\mathbb{P}$-matrix. A map $\delta:\operatorname{M}(r,R)\rightarrow\mathbb{R}$ is determinant-like if it satisfies the conditions of the paper's stated lemma. Cauchy–Binet conjecture. If $\delta$ is determinant-like, then

$$\delta(AA^\dagger)=\sum_{B\subseteq E:\,|B|=r}\delta\bigl(A[X,B]A[X,B]^\dagger\bigr).$$

This is proposed as a generalization of the Cauchy–Binet identity beyond skew fields, contingent on extending the proof to weak matrices over skew partial fields.