Proceedings began on Monday at the UK's highest court as the British government attempts to overturn a ruling that Parliament must hold a vote before Britain's European Union exit negotiations can begin.
Prime Minister Theresa May plans to trigger two years of divorce talks before the end of March by invoking Article 50 of the EU's key treaty, using centuries-old government powers known as royal prerogative.
The powers - traditionally held by the monarch but now used by politicians - enable decisions about joining or leaving international treaties to be made without a parliamentary vote.
Financial entrepreneur Gina Miller and hairdresser Deir Dos Santos went to court to argue that leaving the EU will remove some of their rights, including free movement within the bloc, and that shouldn't be done without Parliament's approval.
Last month, three High Court judges agreed.
But the government says they have misinterpreted the law.
Government lawyers led by Attorney General Jeremy Wright will argue this week that the British people have spoken by voting in a June 23 referendum to leave the EU.
They say a parliamentary vote would mean getting lawmakers "to answer precisely the same question which was put by Parliament to the electorate and has been answered in the referendum."
Many legal experts say the government will likely lose its appeal and be forced to give Parliament a vote.
In a reflection of the constitutional importance of the case - which hinges on the balance of power between Parliament and the executive - all 11 Supreme Court judges will hear the appeal, the first time the full court has sat since it was founded in 2009.
The case is scheduled to last four days, with a ruling likely in January.