Indeed: I mentioned his victory on the battlefield, defeating his Yorkist rival, King Richard III (whose niece, Elizabeth, he would later marry) -- the SINE QUA NON of his claim on the throne. For if I understand correctly, he had virtually no claim whatsoever on the basis of genealogy, since (a) he descended from King Edward III through the Lancasters (whom subsequent history would deem as having a weaker claim than the Yorks, whose claim was validated by the order of male-preferred primogeniture: the precedent for the succession passing through females had been established earlier on, with the Stephen vs. Matilda controversy); and (b) more importantly, it was through an illegitimate line.
Of course, it helped that by then, those with superior genealogical claims (regardless of which house they belonged to) had largely killed one another off, during the War of the Roses. It also helped that his claim on the English throne was supported by the nobles. I've never heard of any historian validating his claim on some kind of hereditary right.
But while his own claim was based on pure conquest (by the sword), his children's claim had a basis in genealogy -- since their mother was Elizabeth of York.
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