The Laundry:—Thos. Noryce; five others. No other mention of laundry or launderers/laundresses on the linked page.
Mention of a [lady] Rice among the Queen's attendants (knights' wives), no specific details. No reference to laundresses.
Sir "Rice" ap Thomas present, check. Sir Griffith ap Rice present, mentioned twice. No "5,000 horses" - there were not that many in the entire English delegation.
The page does not say what you claim it says.
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As to Henry Carey, your link is once again malformed. However, he and his sister are *well* known to have been possible children of Henry VIII and were *certainly* children of his former mistress Mary Boleyn (sister of Anne). Old news.
As to Anne of Cleves, she was married to Henry VIII at one time, but it is widely believed that the marriage was never consummated (if I recall, she had to agree to that in order to get the marriage annulled). So big whoop.