Looked at, probably an updated version/description of the adapter, both says supports analog audio out but not video.
'This adapter lets you connect devices with a Lightning connector to many of your 30-pin accessories.* Support for analog audio output, USB audio, as well as syncing and charging. Video output not supported.'
So from what i can tell. Audio would work as normal, just no video.
PS: Lightning and Thunderbolt is Apples name for their new connectors. And Lightning is supposedly faster and all digital.