Corollary 3

Suppose the $E_i$ are complete and $\phi(E)$ is closed in $F = \prod_{i \in I} E_i$. Then E is complete in the topology $\mathscr{T}$.

For the subspace $\phi(E)$ of F is then complete (GT, II, § 3.4, prop. 8 and § 3.5, prop. 10), therefore the same is true of E in the inverse image topology (GT, I, § 7.6, prop. 10 and GT, II, § 3.1, prop. 4).