Hi branchq,
Yes, perhaps where we have been posting at cross-purposes:
sAs a matter of practice, the Sovereign usually grants each son a dukedom with ubsidiary peerages upon marriage in order to provide a new family line and style for their descendants, who lose the right to carry the prefix Royal Highness after the first generation, but continue to hold a royal peerage through the eldest male with the style of a peer.
They are then addressed by their ducal style "HRH the Duke of York", which is their title, but remain "HRH the Prince Andrew" as well. When they marry, their wives become princesses of the UK with the style of HRH and addressed by their peerage titles
My concern here has not been with princely rights, for I assure you I fully agree with you about the dignity of Prince of the UK, Where we are disagreeing is, I think, on the title or style. The child of the Sovererign under letters Patent has the right to be a Prince(ss) of the UK with the style HRH. Where you and I apparenly disagree, that is if I am reading you correctly which I might not be, is that you are interpreting that grant to read that the HRH/Princely style must include the Christian names of the Prince(ss) in question.
On the contrary it seems from the BRF website that Christian names are only used with the Prince(ss)s style where no higher dignity applies.
It seems that the BRF website very explicitly states in all cases that once the higher title was granted, the style that used the Christian name was retired: Andrew ceased to be HRH The Prince Andrew, certainly it is explicitly stated that Anne ceaed to be HRH The Princess Anne, and Edward ceased to be HRH The Prince Edward.
Thier rights as HRHs and Prince(ss)es of the blood royal under the 1917 letters are not affected. Where we are disagreeing is over the title. the HRH + Christian name ceased to be used, and became incorrect to be used, where a royal ducal or royal earoal (?) dignity, a higher dignity, had been put in its place. That is why we have what we have in the BRF website for the Countess of Wessex, see above. She was not The Princess Edward because he had become HRH The Earl of Wessex, thus, as it says explicitly, she became HRH The Countess of Wessex, and *not* HRH The Princess Edward.
What all this proves is that there is no title HRH The Prince Andrew, Duke of York (and subsquently HRH The Princess Andrew) but that the title is only, and solely, HRH The Duke of York.
HRH The Prince Andrew, Duke of York is not a real title, is not official, and actually does not exist; the Prince in question bears the official title HRH The Duke of York and no other (unless we are talking subsidiary titles).