Well i may have misunderstood, but in that case, your argument does nothing to support your case and in fact doesn't defeat mine... The dwarves are the same 'species' aka, the same difference as between a black man and a white man. I will suggest you read my earlier example once more.
Edit: Furthermore, as I said earlier, there is no indication of any rivalry of any sort within the settings, though by their psychological natures they would seem to have maybe a sporting rivalry, friendly rivalry [sibling rivalry ;-)], but as for them disliking each other, totally wrong. I would expect that dwarves would stand side by side as both are 'shorter'. Calling each other 'brother' would suggest a sense of brotherhood between them due to their many similar traits.