How can you prove that the web server (iis) role is installed on the server?

I'm sure the answer is not what you are looking for, but from many years of personal experience the simplest way is to drop to a command prompt and do a net start - this will give you a list of all the running services. Alternatively, you can run WIN+X and run services.msc and check if the World Wide Web Service is there and running. You can also try to connect to the server via its server name, If IIS is installed the Default page should at least come up, otherwise you might get a 404 or 500 error message indicating something is wrong with the setup.
There are many more way to do this but hopefully this will suffice.