Five Supreme Court justices are analysing an alimony row between a divorced couple whose marriage ended 16 years ago.
Graham and Maria Mills have already staged fights in a family court and in the Court of Appeal.
Their dispute has centred on the size of maintenance payments Mrs Mills gets from Mr Mills.
Supreme Court justices are focusing on arguments about alimony paid to cover Mrs Mills' rent.
Lady Hale, the Supreme Court President and most senior justice, said, at the start of the hearing on Wednesday, that the case raised a point of law of "some considerable importance".
Judges have heard how the pair, who are in their 50s and have links to the London area, divorced in 2002 and agreed that Mrs Mills should get a lump sum plus monthly payments of £1,100.
Four years ago Mr Mills, a surveyor, returned to a family court and argued that the payments should be reduced.
He said Mrs Mills had lost the lump through "gross financial mismanagement" and argued that she was in a position to work and increase her earnings.
Mrs Mills disagreed.
A family court judge ruled that nothing should change but Court of Appeal judges, who considered the case at a hearing in London in 2017, said Mrs Mills' payments should rise by £341 to £1,441 a month.
Mr Mills wants Supreme Court judges to overturn decisions made by Court of Appeal judges.