Well its not fine. I think what you are trying to say is that all ales and largers are beer, but not all beer is ale or lager.

Over here beer, ale and lager are different things. We don't see any american 'Ales' for obvious reasons and most of us think of american 'beer' as budwieser or miller.

Our 'Ales' are, generally, living beers, though there are many different types - including both bottled condituioned ale and 'sterile' bottled ales. There are bitters, milds, stouts and god knows what else.