We shall use the homological nature of the Solomon functional and the fact that $H_{n+1}(X)=0$. We pick $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ such that

$$\partial\mathcal{C}_{1}=L-L_{0}^{\prime},\quad\partial\mathcal{C}_{2}=L_{0}^{\prime}-L_{0},\quad\partial\mathcal{C}_{3}=L-L_{0}.$$

Then $\mathcal{C}_{1}+\mathcal{C}_{2}$ is homologous to $\mathcal{C}_{3}$, so we can replace $\mathcal{C}_{3}$ by $\mathcal{C}_{1}+\mathcal{C}_{2}$ to compute $\mathcal{S}_{L_{0}}(L)$, whence (22) follows. ∎