Proof of the main covariance statements

The off-shell statement (iv) uses only coordinate covariance and the fixed coordinate data; it is valid for the larger algebra of fields supported in any $U^{\prime }$ with $\overline{U^{\prime }}\subset \{\epsilon <t<T\}$. The on-shell statement (v) additionally uses $\overline{U}\subset P_g$. The theorem restricts to any ${\mathcal{G}}_U$-invariant subset of configurations; invariance of a given subset, such as the Einstein–scalar jet of VII-A, is a separate statement about the metric equations and is not reproved here.