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They weren't just "negotiating" a marriage between them: secret marriages were not uncommon back then. Edward IV, one of the "negotiators" had been married secretly and so had Queen Katherine of Valois, mother of the Tudor family. The marriage was secret because it was planned that the couple would claim to be king & queen of Cyprus.
Anthony Rivers was married three times: 1st to Elizabeth Scales, 2nd to the "half Beaufort" Mary Lewes, and 3rdly to Princess Margaret Cecilia Stewart of Scotland. He had had an illegitimate child sometimes erroneously described as legitimate much earlier with Gertrude Stradling (whom he never married) called Margaret Woodville who was specifically excluded from using the name "Rivers". But his only legitimate child, born in wedlock, was the child of Princess Margaret Cecilia Stewart of Scotland, the daughter of King James Stewart II (ruled 1437-60) of Scotland called "of the fiery face" because of a huge vermillion birthmark and his wife Marie of Guelders, and the granddaughter of Katherine Princess of Kleve and Arnold vanGelre Duke of Guilders or Guelders in the Netherlands. She was born about 1459 in Stirling, Stirlingshire, Scotland and died 1493. Just as Anthony's affair with Gertrude Stradling is often erroneously described as a marriage, so is Princess Margaret's marital status often erroneously reported. She is said to have had "an affair" with Wm Crichton 3rd Lord of Crichton from which an illegitimate child issued, but she was properly married to Lord Crichton by that time. Anthony had died in 1483 and Princess Margaret's daughter with Lord Crichton, Margaret Crichton, was first married in 1507 (to Wm Jodrick of Edin, 2nd to Geo Halkerstoun who died 1513, and 3rdly in 1513 to Geo Leslie 2nd Earl of Rothes) so she could have been 24 by the time of her 1st marriage even if she had been conceived within a year of her mother being widowed.