Okay, they've given you that the maps and are continuous maps from . A property of the product topology is that a composite function is continuous if and only if its component functions are. Use the fact that a composition of continuous functions is continuous and you're done.
Edit: That might not be clear. The point is that the map is a composition of two maps, where and , where the "s" stands for "sum", and . They've told you s is continuous, and you know F is continuous since its components are continuous and its codomain has the product topology. Therefore the composition is continuous.