BBC legal correspondent
This was a case about a hugely important point of pure constitutional law, but it arrived at the Supreme court in a blizzard of politics, acrimony, threats of violence against Gina Miller the woman at its heart, and some very personal press criticism of the judges about to hear it.
Like many Supreme Court cases it rapidly took on the feel of an academic seminar, often inaccessible and at times impenetrable to the non-lawyer.
Not a ratings winner then, but it has been a fine example of the rule of law and the British constitution in action. Independent judges considering issues raised by citizens, exercising their right to have a decision of ministers scrutinised by a court to determine whether it is lawful or not.
The proceedings have been unfailingly courteous and all of the key players have had their arguments carefully considered. That is not something that happens in every country around the world. So, even if the nation remains divided on the issue, it can take pride in the process by which it is being determined.Quote Message:
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Gina Miller, the woman behind the Brexit legal challenge, said Wednesday night's vote on the issue "simply won't suffice".
MPs voted to back the government's plan to start formal talks on Brexit by the end of March next year.
They also supported a Labour motion calling for Parliament to "properly scrutinise" the government in its proposals for leaving the EU.
But in a statement after the Supreme Court hearings, Ms Miller, an investment manager and philanthropist, said:Quote Message: "I hope that the Supreme Court will uphold the High Court ruling that Article 50 cannot be triggered using the royal prerogative. Our case is that prerogative powers end where domestic law begins. Only Parliament can grant rights and only Parliament can frustrate, nullify or displace rights."
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Giving a statement after the close of the Supreme Court hearings, Attorney General Jeremy Wright argues that the government had the power of the Royal prerogative to trigger Article 50.
"Parliament could have at various points said: 'We don't want the executive to trigger Article 50 to start this process of leaving the EU without parliamentary sanction' - and on all of those occasions it did not say so," he said.
"So that leaves, we say, the power in the hands of government ... to do what the people have said they want doing."
BBC News Channel
Outside the Supreme Court, Robert Pigney, from the People's Challenge, explains his position, saying he believes MPs should be in control of the decision to trigger Article 50.
"We believe it's not within the prime minister's authority to trigger Article 50," he says.
Barrister Jeremy Brier, from Essex Court Chambers, says if he was on the government's side "I wouldn't be feeling terribly confident".
He claims James Eadie QC opened the case for the government "confidently" on Monday.
But he says "the hostility" from some of the Supreme Court justices "seemed to suggest a lot of disagreement".
Robert Craig, from the London School of Economics, conceded that questions from some of the judges had been "sharper and harsher".
But he argues that even if the government loses the Supreme Court case, "there's no way that could delay Brexit" because the government could attempt to trigger Article 50 through Parliament.
Supreme Court president Lord Neuberger thanked "everyone involved" in the presenting their case and for the work behind the scenes.
"In addition to counsel, we're also grateful to all those inside and outside the court who have been converting our legal discussions into a more accessible form for members of the public," he said.
In his closing remarks he said:Quote Message: We are not being asked to overturn the result of the EU referendum. The ultimate question in this case concerns the process by which that result can lawfully be brought into effect. As we have heard, that question raises important constitutional issues and we'll now take time to ensure the many arguments which have been presented to us orally and in writing are given full and proper consideration. We appreciate this case should be resolved as quickly as possible and we'll do our best to achieve that."
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James Eadie QC, for the government, says it's "highly significant" though not legally binding that MPs voted on Wednesday to back the government's plans to start formal talks on Brexit by the end of March next year.
In his closing remarks, he added that this is a matter the court can take into account.
James Eadie QC, for the government, insists that Article 50 is about withdrawal from the EU.
"We submit that's significant. The Royal prerogative powers which the 1972 act have done nothing to take away, remain," he said.
"It isn't a statutory power - Article 50 - but it involves Parliament and legislation recognising its existence and acknowledging its effect
"And all of that is critical... and what it tells one about Parliament's intention on the division of constitutional responsibility in relation to Article 50. This was the very mechanism by which the very thing which is now challenged was to be done and Parliament decided, as it did, no control."
He added:Quote Message: It's absolutely obvious that if Article 50 notice is given, the process of withdrawal is commenced, the bullet is fired at the target, with all the potential effects that has on directly effected rights and obligations... The idea that Parliament didn't know that was the effect of Article 50 couldn't be sustained."
James Eadie QC, for the government, says Parliament enacted the 1972 European Communities Act (ECA)
"I have to show that the nature of the parliamentary intervention in this context from 1972 onwards allows the government to continue to exercise its prerogative powers on the international plain..." he says.
"You don't freeze the clock in 1972 - you look at the statutory scheme as a whole and you make your mind up."
He adds that the ECA is legislation designed to implement UK treaty obligations - "it wasn't seeking to control the prerogative, in other words the action by government on the international plain".
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Responding, as we get close to the end of the four day hearing, the government's lawyer James Eadie QC says it does not plan to alter the law of the land by using the Royal prerogative - adding that EU law isn't the law of the land.
If the prerogative power to conduct UK foreign affairs cannot be exercised, it has "very serious consequences", he says.
BBC Wales Parliamentary correspondent David Cornock reports from the Supreme Court on the Welsh Government's intervention in the Brexit hearing.
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Patrick Green QC opens the afternoon session by saying he is speaking on behalf of the "Expat Interveners" - George Birnie and others - who will be affected by key rights that may be lost as a result of the triggering of Article 50.
He argues that the conferral of legislative power on institutions and investing rights on individuals could only be exercised by Parliament.
The World at One
BBC Radio 4
Former foreign office minister Alistair Burt told BBC Radio 4's The World At One that he wouldn't have made critical comments about Saudi Arabia in public.
But in private it can be quite different - "making sure our relationships are strong does not preclude being very direct with people when we need to be," he added.
Radio 4's Westminster Hour
In an extended version of his interview with the Westminster Hour's Sam Macrory, historian David Starkey, who was a member of the Brexit-supporting Historians for Britain, discusses the result of the referendum and what it means for Britain and the future of the EU.
Dr Starkey, also explains why he found the verdict of the High Court on Article 50 to be "disgraceful", why those who fear that Europe is returning to the 1930s are "silly", and explains how the actions of Henry VIII helped to shape England's relationship with the rest of Europe today.