Proof

If Φai, Φbi, ⋯, Φni are the incoming fluxes from other tanks of the system, ΦiA, ΦiB, ⋯, ΦiJ are the outgoing fluxes, and Φ<sup>1</sup> <sup>0</sup>, Φ<sup>2</sup> <sup>0</sup>, ⋯, Φ<sup>s</sup> <sup>0</sup> are the incoming fluxes from outside the system, then the corresponding ODE can be written as

$$V\_i \frac{d\mathbb{C}\_i}{dt} = \Phi\_{i\ell}\mathbb{C}\_{\mathfrak{t}} + \dots + \Phi\_{\mathfrak{m}}\mathbb{C}\_{\mathfrak{n}} - \left(\Phi\_{i\mathcal{A}} + \dots + \Phi\_{\vec{\ell}}\right)\mathbb{C}\_{\mathfrak{i}} + \Phi\_0^1 \mathbb{C}\_0 + \dots + \Phi\_0^s \mathbb{C}\_s \tag{20}$$

This equation gives.

$$\frac{d\mathbf{C}\_i}{dt} = \frac{\mathbf{Q}\_{ai}}{V\_i}\mathbf{C}\_a + \dots + \frac{\mathbf{Q}\_{ni}}{V\_i}\mathbf{C}\_n - \frac{\sum \mathbf{Q}\_{ij}}{V\_i}\mathbf{C}\_i + \frac{\sum \mathbf{Q}\_0^p}{V\_i}\mathbf{C}\_p\tag{21}$$

Eq. (20) implies that the ith row of the MP-matrix has entries: <sup>Φ</sup>ki Vi for k 6¼ i, � P<sup>Φ</sup>ij Vi for k ¼ i, and PΦ<sup>p</sup> 0 Vi Cp is part of the independent term.

A flux balance gives <sup>P</sup>Φki <sup>þ</sup> <sup>P</sup>Φ<sup>p</sup> <sup>0</sup> <sup>¼</sup> <sup>P</sup>Φij, which implies <sup>P</sup>Φki <sup>&</sup>lt; <sup>P</sup>Φij, and then: aii ¼ � P<sup>Φ</sup>ij Vi < 0 and also Ri ¼ P<sup>Φ</sup>ki Vi < P<sup>Φ</sup>ij Vi ¼ aii j j, which proves the theorem.
