Suppose that neither is a subset of the other.

This means

What is meant here is that there is an element 'a' which is in U but not in V

Similarly there is an element 'b' which is in V but not in U

This statement follows directly from the fact that U is not a subset of V and V is not a subset of U

If

is a subspace then

both 'a' and 'b' (defined above) will belong to

which we assume to be a sub-space. As subspace is closed under addition so a+b =c will also belong to the union of U and V.

But if

then

or

That is a contradiction. Similarly if

we get a contradiction.

Thus one of U or V is a subset of the other.

As c belong to the union it has to belong to (at least) one of U,V. Let the chosen one be U. Now c and a both belong to U. Thus c-a=b will again below to U (under the closure and inverse axiom). But we always maintained b doesn't belong to U. Hence contradiction !!

Same can be argued for V.

Hence the result. Thanks