The High Court dramatically ruled last month that Prime Minister May's government did not have the power to invoke Article 50 of the EU's Lisbon Treaty
The High Court dramatically ruled last month that Prime Minister Theresa May's government did not have the power to invoke Article 50 of the European Union's Lisbon Treaty, the formal procedure for leaving the EU.
The judgement prompted fury amongst Brexit supporters who fear that lawmakers, who are overwhelmingly in favour of staying in the EU, may seek to delay or soften Britain's withdrawal.
They have warned of a potential "constitutional crisis" as the judges rule on the limits of executive power.
Following a heated and divisive campaign, Britons voted by 52 percent to leave the EU in the June 23 referendum.
But the act legislating the vote did not make the result legally-binding, meaning either the government or parliament still has to pull the trigger.
In the shadow of the Houses of Parliament, all 11 Supreme Court judges will on Monday begin four days of appeal hearings, with a decision due in January.
Despite the complexity of the issues involved, they will be under pressure to make a swift ruling, as May has promised EU leaders she will invoke Article 50 by the end of March.
May argues that as head of the government she has constitutional authority over foreign affairs, including the right to withdraw from treaties, under so-called "royal prerogative" powers.
But the claimants in the case, led by investment fund manager Gina Miller, counter that Brexit would nullify some domestic laws and strip citizens of certain rights -- actions that only parliament can carry out.
The High Court ruling against the government was cheered by opponents of Brexit, who hope that pro-European lawmakers may be able to use a parliamentary vote to ease the terms of the divorce, for example by keeping Britain in the single market.
But the decision prompted personal attacks on the judges from members of May's Conservative party and in the eurosceptic media, with one tabloid calling them "Enemies of the People".
An added complication in next week's hearings will be the presence of representatives from the devolved Scottish and Welsh governments, who are expected to argue that Article 50 also needs to be approved by their devolved parliaments.
Such a ruling could derail May's timetable further and, given that Scottish lawmakers are opposed to leaving the EU, set up a stand-off between the nations.
The Supreme Court will also hear an appeal calling for the Northern Ireland assembly to have a vote, brought by Raymond McCord, a victims rights campaigner.
He is concerned that Brexit may result in Britain withdrawing from the European Convention of Human Rights, which he fears would undermine his fight for justice for his murdered son.
While the government has publicly expressed confidence about its appeal, legal expert Michael Zander said it has little chance of winning, describing the original ruling was "unanimous and very strong".
"In my view, the government could be looking at losing 11-0," he wrote in legal magazine Counsel.
If it does lose, the government is expected to immediately introduce a short bill authorising the invoking of Article 50 that it will try to rapidly push through parliament.
The main opposition Labour party, which has 231 MPs in the 650-seat House, has said it will not block Article 50 but it is divided on the issue.
The government is also braced for a potentially complex judgement, with Brexit minister David Davis this week telling MPs that "it isn't just a yes-no outcome".
Supreme Court judge Brenda Hale, one of those hearing the appeal, suggested during a recent speech that the 1972 European Communities Act which is the foundation of Britain's EU membership may have to be entirely replaced before Brexit could begin -- a process that would bring even further delays.
Conservative MP Iain Duncan Smith, a leading Brexit campaigner, said that would cause "a constitutional crisis", and warned that Hale held pro-EU views.
"It is not their job to tell parliament... how they should go about that business, that's for parliament to decide," he told the Guardian newspaper.