"@ cteno4: Granted I do accept evolution and natural selection as a more-or-less correct understanding of life on this planet, but I have hard time believing that there's mathematical proof of it (I'm not calling you a liar, or anything. I actually find this stuff facinating and would love you to explain it). Also it's not clear whether organisms like bacteria develop immunities through and evolutionary process. Even in TB cases, an individual infection can become immune to it's treatment if the patient stops medicating for even a few weeks. Most evolutionary changes, even on a microscopic level, occur over millennia, not weeks. So it's not really certain whether it was this process that lead to the bacteria's development or a separate chemical alteration within them."
@Tetra. I wouldn't say evolution is so much mathematically proved as it is overwhelmingly empirically proven. There are however mathematical models that help describe evolution though they don't really prove the theory as much as they ride upon it as useful tools when modelling evolution.
Evolution is based on a few premises (not necessarily in this order) which have been empirically proven and natural selection follows as the logical conclusion.
1. There are finite resources and more individuals than can necessarily be supported, thus competition between the individuals.
(True, this is a unanimous theme in nature.)
2. Individuals are not identical (True, genetic variation in the form of mutations, sexual reproduction, and crossover via meiosis)
3. The more well adapted individuals are more likely to acquire the finite resources and reproduce and less likely to die. (Keyword "likely", luck plays into this but this is the general tendency yes?)
4. The characteristics that make these individuals more well adapted are heritable. (True, that mechanism being DNA)
It's not hard to see that it logically follows that genes which express themselves as favourable phenotypes in an individual are naturally selected for success in the gene pool and more and more individuals in the future will likely have this gene.
Regarding microbial evolution, the process that confers immunity is almost (barring a very rare exception?) certainly evolution. The reason it happens relatively fast is that bacteria have extremely short generations and because evolution acts between generations and not during an individual lifespan this means that bacteria evolve much faster than organisms with longer generations. This is why bacteria are by FAR the most populous and most diverse group of organisms on the Earth.
As well, antibiotics, being designed to take out these bacteria represent a very serious change in the bacteria's environment. Since the antibiotics represents the major selecting influence upon the bacteria, any bacteria with even a marginal advantage or resistance to the antibiotics will rapidly be selected for, and this process continues very quickly because evolution runs on a positive feedback loop. If you take the immune bacterial culture outside of the antibiotic, you find that the cost of maintaining resistance to an antibiotic that is no longer present penalizes those bacteria that are unnecessarily immune and the culture reverts to it's previous state of vulnerability to antibiotics.
Evolution only takes place on a longer span of time for organisms with longer periods between generations or during long periods in which the environment remains relatively static.