Corollary 4

Let X be a topological space, Y a uniform space, H an equicontinuous set of mappings of X into Y. If H is endowed with the topology of pointwise convergence, then the mapping $(h, x) \to h(x)$ of $H \times X$ into Y is continuous.

More intuitively, this expresses the fact that if $h \in H$ converges pointwise to $h_0 \in H$ and if $x \in X$ converges to $x_0$, then $h(x)$ converges to $h_0(x_0)$.