1. Do you accept that there exist complex molecules in the cells of living organisms which are known as "DNA?"
Yes.
2. Do you accept that, within these DNA, there are individual "genes?"
Yes.
3. Do you accept that there are naturally occuring factors, like radiation and toxins, that can alter the genes within a given cell?
Yes.
4. Do you accept that cells reproduce themselves within the organism?
Yes.
5. Do you accept that the genes of a cell that is reproducing will be duplicated to the new cell, even if changed by an external influence?
For the most part, if I understand the question correctly.
6. Do you accept that an offspring's genes are determined by the genes in the reproductive cells of both parents that were used to concieve the progeny?
Yes.
7. Do you accept that any changes that were made to the genes of one of the above reproductive cells will be passed on to the progeny?
Yes.
8. Do you accept that such changes can result in differences in how the organism appears, its capabilities, or to its level of intelligence?
Yes.
9. Do you accept that, if a change is entirely beneficial, the organism has a higher chance of surviving than the rest of its species?
For the most part. Some "Bad" traits are passed on from time to time. Some lessor adapted individuals still get the chance to procreate.
10. Do you accept that an organism that has a higher chance of surviving has a higher chance of reaching the age of sexual maturity?
Yes.
11. Do you accept that, if an organism has a higher chance of reaching sexual maturity, it has a higher chance of reproducing? (remember, we are assuming that said genetic difference is entirely beneficial)
Yes.
12. Do you accept that, as in the case with the organism's parents, it will pass on its genetic variation to its offspring?
Yes.
13. Do you accept that, if the organism produces progeny which have the same genetic difference, which has already been established as beneficial, that it is likely that this variation will eventually encompass greater and greater numbers of the species, until the entire population has changed (though not necessarily the entire species, as there may be geographical barriers separating the variation from mating with other populations of the same species)?
Yes.
14. Do you accept that, eventually, a population of organisms may become so radically different from the original that they lose the ability to reproduce with other populations of the original species (if any exist)?
Possibly. Points to Neanderthal, and Homo Sapien. (before Homo Sapien became Homo Sapien Sapien)
15. Do you accept that, given that scientific classification of organisms is based on genetic differences, the above population of organisms is a new species?
Somewhat. I guess we'd call them brothers, or possibly cousins.