There are several premises in this logical progression which could alter it in a very profound manner. There are two premises which stick out to me as being needed for this logical progression to be defended.
First, although there is a statement claiming pain is a bad thing, it is not listed as a premise. It is at best a quasi-premise. To withstand an attack on the logic, it would need to state with certainty that "Pain is a bad thing". In this example, my guess is the opening statement is meant to serve as a premise.
The missing premise is therefore located further along the logical progression. Why is it clear we should not eat meat? The logical progression is attempting to argue people should not eat meat to avoid causing animals pain. A noble thought, but not logically sound in this example. If we do not eat meat, then what do we eat? The answer is obvious in that people do not need to eat meat to survive. However, in logical progression nothing can be assumed. Therefore the missing premise is something to the effect: "People do not need to eat meat to survive".
Remember, when dealing the logical progressions everything must be presented as a premise (which is really just an assumption you're basing your logical progression on). The reader cannot be expected to assume anything because people will assume all manner of variations not covered by the logical progression.