Lemma: Stability of the equation ideal

The derivation $s_0$ preserves the free equation ideal.

Proof. Since $A\mathcal{R}=0$ and $s_{0}b=0$,

Also,

The ghost equation for $c$ is unchanged, while

Taking the divergence of the metric–auxiliary equation gives $\square b=0$: the divergence of $A\gamma$ is zero and the divergence of $2D^{\ast }b$ is $-\square b$. The scalar equation is unchanged. Thus every transformed equation belongs to the same linear equation ideal.