Let f, g be linear mappings. We consider the function f ∘ g, defined as
f ∘ g = x ↦ f(g(x).
Firstly, we know that
f(g(x + y)) = f(g(x) + g(y))
since g is a linear mapping by assumption. Now we use the fact that f is a linear mapping to conclude that
f(g(x) + g(y)) = f(g(x)) + f(g(y)).
We have therefore shown that (f ∘ g)(x + y) = (f ∘ g)(x) + (f ∘ g)(y) and so we have established the first linear mapping law.
The second part of the proof is very similar: we show that f ∘ g is compatible with scalar multiplication by first using the fact that g is compatible with scalar multiplication and then by using the fact that f is.