Bon chance, Charlie

I found the following information in a book of Ionia County, Michigan: ".....Thomas and Jane (DeYarmond) Spencer, both natives of Nova Scotia, Canada, the former of whom was the grandson of an Irish immigrant who settled at Halifax and the latter of French and Scottish parentage, her father having been of French descent and her mother, who was a Fletcher, having been of Scottish descent. In 1831, Jane DeYarmond accompanied her father on a trip to the Territory of Michigan and was at Detroit for a year, at the end of which time they returned to Nova Scotia and presently returned to Michigan with a party which settled in Washtenaw county, not far from Ann Arbor. One of the members of this party was Thomas Spencer and not long after their arrival in this state he and Jane DeYarmond were united in marriage......."

I also found this "Alexander, the third son of Alexander Deyarmond and Mary Barnhill, was born about the year 1766. He was married to Mary, daughter of the late Thomas Fletcher, of Masstown. They had four sons and five daughters."

I need to know if Jane really is the daughter of Alexander DeYarmond and Mary Fletcher.