A Riemannian product is complete iff each factor is complete

Suppose first that $r>0$. By [F1], finite choice gives a point of $P$, so $P$ is nonempty; [F1] also makes it connected. Repeated product charts take values in ${\mathbb{R}}^{n_0}\times \cdots \times {\mathbb{R}}^{n_{r-1}}={\mathbb{R}}^{n_0+\cdots +n_{r-1}}$, so the factors' boundaryless local models make $P$ boundaryless. Thus [F5] applies both to $P$ and to every factor.

Write a product coordinate as $x=(x_a^i)$. By [F2], entries of the $a$-th diagonal block are the coefficients of $g_a$ and depend only on $x_a$; all off-diagonal entries vanish, and the inverse matrix has the corresponding inverse diagonal blocks. Substitution in [F3] shows that a Christoffel symbol with all three indices in the $a$-th block is the corresponding symbol of $g_a$, while every symbol involving more than one block is zero. Indeed, in each term of the Christoffel formula either a metric entry is off-diagonal or a derivative is taken in a coordinate belonging to a different factor.

If $r=0$, conditions 1 and 2 are vacuous. The product $P$ is the stipulated one-point manifold: its metric is the zero metric on a singleton and hence is complete, and its only initial tangent vector is zero, whose maximal geodesic is constant on $\mathbb{R}$ by [F4]. Thus conditions 3 and 4 hold as well.

Hence, for a smooth curve $\gamma =({\gamma }_a):I\to P$, the coordinate geodesic equation in the $a$-th block is exactly the coordinate geodesic equation for ${\gamma }_a$ in $M_a$. Applying the two directions of [F3] on product-chart subintervals proves $\gamma \text{ is a geodesic in }P⟺{\gamma }_a\text{ is a geodesic in }{M}_a\text{ for every }a<r,$ with the same affine parameter.

Assume condition 2 and fix $(p,v)\in TP$, with components $(p_a,v_a)\in T{M}_a$. By [F4], each factor's maximal geodesic with this initial data is defined on $\mathbb{R}$. Their finite product $\gamma (t)=({\gamma }_a(t){)}_{a<r}$ is smooth and is a product geodesic by step 2.1. It has initial data $(p,v)$; maximal-geodesic uniqueness in [F4] therefore forces the maximal product geodesic to have domain $\mathbb{R}$. Thus condition 4 holds.

Conversely assume condition 4, fix $a<r$ and $(p_a,v_a)\in T{M}_a$. By finite choice in [F1], select one basepoint $p_b\in M_b$ for each $b\mathrm{\ne }a$, and form product initial data with $a$-component $v_a$ and all other velocity components zero. Its maximal product geodesic is global by condition 4. Step 2.1 makes its $a$-th projection a geodesic on $\mathbb{R}$ with initial data $(p_a,v_a)$, so uniqueness and maximality in [F4] make the factor's maximal geodesic global. Since $a$ and its initial data were arbitrary, condition 2 holds.

By step 1.1 and [F5], condition 1 is equivalent to condition 2 factor by factor, and condition 3 is equivalent to condition 4 for $P$. Steps 3.1 and 3.2 give condition 2 if and only if condition 4. Combining these equivalences proves all four conditions equivalent when $r>0$. Applying [F5] separately to each supplied factor is universal reasoning and makes no simultaneous choice of geodesics or witnesses.

The proof includes $r=1$, zero-dimensional factors, zero initial vectors and both infinite-time directions. Nonemptiness of every factor is essential: if one factor were empty, the product would be empty and hence complete vacuously even if another factor were incomplete. Parameter intervals have no finite endpoints after completeness because their domains are all of $\mathbb{R}$. The only non-ZF assumption is [A1], used exactly through [F4] and [F5]; the finitely many auxiliary basepoints in step 3.2 are supplied by the ZF theorem [F1].