What is the Law of Comparative Advantage?

I know in vague terms what the law says. If there are two nations producing goods, even if one can produce everything more efficiently than the other, there is still something that second nation can produce better in comparison to everything else that it produces and to achieve maximum productivity between the two nations that commodity is what the second nation should concentrate on.

I have searched the Web for something a little clearer but have not found anything. Here is how I have reasoned on this. Suppose we have commodities x1, x2, x3 and x4. Let f(x) be the productivity for nation 1 and g(x) be the productivity for nation 2. It is not hard to show mathematically that there is some x among x1, x2, x3 and x4 such that f(x)/f(xi) >= g(x)/g(xi), where xi ranges from x1,..,x4. I chose 4 to be concrete, but the same reasoning applies to any number of goods. Is this what is meant by comparative advantage?

One problem that I have with this is that it does not work for more than two nations. If there is a third nation with productivity function h(x), then, from the above, there is some y such that f(y)/f(xi) >= h(y)/h(xi). The problem is that the x and y are not necessarily the same. The commodity that gives the greatest advantage against one nation would not necessarily be the same commodity that gives the greatest advantage over the other.