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To get back to the original question of Anthony Antioch-Lusigan-Rivers however, the Greek blood had become dilute again. Alice de Champagne (mostly Greek) m. Hugh I de Lusignan King of Cyprus;their daughter Marie de Lusignan m. Gauthier IV comte de Brienne--notwithstanding deBrienne having been a widower of a Queen of Jerusalem and later elected emperour of Constantinople, he was a Roman Catholic and had remarried to a Spannish princess, and was anyway originally French;their son Hugues Comte de Brienne m. Isabelle de la Roche;their son Gauthier V C de Brienne was Duc d'Athene (D of Athens) but, again! a Roman Catholic "Crusader" Duke who m. Jeanne de Chatillon;their daughter Isabelle de Brienne m. Gauthier III d'Enghien;their son Louis d'Enghien, C de Brienne m. Jeanne de St.Severino;their dau. Marguerite d'Enghien m. Jean de Luxembourg;their son Pierre I de Luxembourg, C de St.Pol m. Marguerite de Baux;their dau. Jacqueline de Luxembourg m. Richard Wydeville, Earl Rivers;their son was this Anthony "Antioch-Lusignan-Rivers" or "Anthony Arnite" whom we treat.
The Greek ancestry of Alice of Champagne was by then rather distant and the emperour of Constantinople and the Duke of Athens from whom he descended in the intervening generations, had not been truly Greek but I would LOVE to hear an argument otherwise!
Certainly "Anthony Antioch-Lusignan-Rivers" or Arnite of Venice had some imperial Byzantine bloodlines, a great many connexions to Cyprus as well through French Crusaders, and certainly he did bear the arms of Lusignan of Cyprus at least on his Garter stall-plate, quartered I believe prominently if not in the 1st Quartering.