In reply to your message. My source for the statement on Mary Borrowes, although it cannot by any means be considered as a primary source, may be based on records no longer available.
Taken from the book The Rainsford Familyby Emily A Buckland, published by Phillips and Probert Ltd1932.
p 243 Descent of Frances Rainsford from Sir Marco (or Marcus )Rainsford, Knt.
This states as a footnote " This pedigree has been most carefully made out by Frederick Turner Rainsford. (1) from wills in Record Office, Dublin. (2) MSS Folio23 Trinity College, Dublin.(3) Office of Arms, Dublin., and further information given by Arthur F J Leveson- Gower, brother in law of the Earl of Liverpool. Jan 7 1908
I would indeed be pleased to have any information on Giles Mee Rainsford that you have as my records are sadly scanty there.