Equations of motion with replacing the Lagrangian by irrep diagrams generating functional

Question

I have read that equations of motion of ghosts is equal to

$$\tag 1 \frac{\delta \Gamma}{\delta \bar{c}^{a}(x)} = -\partial^{\mu}_{x}\frac{\delta \Gamma}{\delta K^{\mu , a}(x)},$$ where $\Gamma = W - \int d^{4}x \sum_{i}(J_{i} \cdot \varphi^{i})$ is generating functional for irreducible Feynman diagramms of nonabelian gauge theory given by $$S = \int \mathrm{d}^{4}x\left( -\frac{1}{4}F_{\mu \nu}^{a}F^{\mu \nu}_{a} - B_{a}f^{a} + \frac{1}{2 \varepsilon}B_{a}B^{a} - \bar{\omega}^{a}M_{ab}\omega^{b} + K_{\mu}^{a}\delta A_{a}^{\mu} + L_{a}\delta \omega^{a} \right),$$ and $K_{\mu}^{a}$ is the source for $\delta A_{\mu}^{a}$, where $\delta \varphi$ is BRST transformations of $\varphi$ field (for other descriptions see here).

Why $(1)$ is correct (i.e. why we can replace $L$ to $\Gamma$ in equations of motion)?

This post imported from StackExchange Physics at 2014-09-01 11:14 (UCT), posted by SE-user Andrew McAddams