A finite straight-line C2 program defines a C2 map on each fixed branch

Each input coordinate map is linear and has constant derivative, so it is $C^2$. Suppose all parent-node maps of operation $j$ are $C^2$. The tuple $g_j$ of parent values is $C^2$, since it is a finite product of their coordinate maps; its domain is open by [F3].

Write $z_j={\varphi }_j\circ g_j$. By [F2], $D{z}_j=(D{\varphi }_j\circ g_j)D{g}_j$. The map $D{\varphi }_j$ is $C^1$ by the $C^2$ hypothesis, and $g_j$ is $C^1$, so $D{\varphi }_j\circ g_j$ is $C^1$ by the chain rule. The derivative map $D{g}_j$ is $C^1$ because $g_j$ is $C^2$. Matrix multiplication is a bounded bilinear map in finite dimensions, so the product rule in [F2] makes $D{z}_j$ $C^1$. Thus $z_j$ is $C^2$ by the definition in [F1].

Induction through the finite operation list gives $C^2$ for every operation value. The scalar output is therefore $C^2$ whether it is one of these values or an input coordinate projection. On a strict branch the selected primitive maps stay fixed on an open neighborhood; no claim is made at branch switches.