By Zhang’s inequality (see [18]), one has

$$e(\overline{L}+\overline{M})\geq\frac{1}{2(\ell+m)}({\widehat{c}}_{1}(\overline{L}+\overline{M})^{2}|X).$$

Since $({\widehat{c}}_{1}(\overline{L})^{2}|X)=({\widehat{c}}_{1}(\overline{M})^{2}|X)=0$ by assumption, we observe that

$$({\widehat{c}}_{1}(\overline{L}+\overline{M})^{2}|X)=2({\widehat{c}}_{1}(\overline{L}){\widehat{c}}_{1}(\overline{M})|X)=-\frac{1}{\ell m}({\widehat{c}}_{1}(m\overline{L}-\ell\overline{M})^{2}|X).$$

This shows the first claim.

Since $mL$ and $\ell M$ have the same degree, viz. $\ell m$, the rest of the proposition follows from the negativity properties of the height recalled above. ∎