Supreme Court case ends with reminder it's not about stopping Brexit
Government appealed against ruling it needs MPs' approval to trigger Brexit
Judgement is expected in January
Watch highlights of each day via clips above, or scroll down to see how events unfolded
The World at One
BBC Radio 4
Labour's shadow Brexit secretary Keir Starmer tells the BBC the High Court ruling on Article 50 was "clear and decisive" and while he could not predict the outcome of the Supreme Court appeal, he suspects that the government might "struggle with its arguments".
Image source, PALord Neuberger has brought proceedings to a close.
He says there will be a change of personnel this afternoon, with Northern Ireland's attorney general John Larkin and Lord Pannick, representing those challenging the government, taking the floor.
But he says he hopes that everyone will still have time to have lunch and be back for a 14:00 GMT restart.
Image source, Supreme CourtIn a bullish final statement before lunch, Lord Keen says that Parliament has had every chance to limit the government's prerogative powers in international affairs but had for "particular reasons" decided to "remain silent".
Court president Lord Neuberger says that the Scottish advocate's argument could be "turned on his head" and that the 1972 European Communities Act - the piece of legislation that paved the way for the UK joining the then European Economic Community - imposed a "clamp" on the government's room for action.
Lord Reed takes the argument on, saying that the "clamp" deriving from the 1972 Act should not be regarded as an "on-off switch".
Lord Keen replies that while it might have been an abuse of power for a government to use prerogative powers to take the UK out of the EU without recourse to Parliament, the fact that Parliament passed a law in 2015 authorising a referendum effectively nullified this argument.
The UK PM was talking to the BBC's John Pienaar
Prime Minister Theresa May says the government wants a "red white and blue Brexit" and insists she is "ambitious" over the deal the UK will strike when leaving the EU.
Image source, Supreme CourtLord Keen is addressing the issue of the Sewel convention, an important component of the 1998 Scottish devolution settlement, which stipulated that Westminster would not normally legislate on devolved matters in Scotland without the consent of the Scottish Parliament.
He suggested that it was a "political" accord and should not be seen as a legal obstacle to the UK government exercising its powers with regard to EU exit without reference to the Scottish Parliament.
Quote MessageThe Sewel convention is a political convention concerning the legislative functions of the Westminster Parliament. It is a self-denying ordinance on behalf of the Parliament. It was never intended to be a justiciable legal principle."
Pressed by Lord Sumption whether he believes that the convention's incorporation into an act of Parliament makes "no legal difference to its effect", Lord Keen replies yes.
EU negotiator Michel Barnier warns the UK that it cannot "cherry pick" the Brexit deal it wants.
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Professor Gavin Phillipson, of Durham University's Law School, admitted this afternoon's argument coming from Scotland was "a long shot" but said the ramifications could be huge.
Quote MessageIf it succeeded it would actually throw a huge spanner in the works. If the Supreme Court was to hold that the Scottish government had to give its consent, we know the Scottish government wouldn’t give its consent. That would actually stop Brexit in its tracks so this is actually an argument that could have huge ramifications if it succeeded."
Professor Gavin Phillipson, Durham University
Image source, Supreme CourtThe session has been going for nearly two and a half hours without a break.
Lord Keen, the UK's chief Scottish legal adviser, apologises for whipping through a series of points "at a rate of knots" without the justices being able to refer to relevant documentation.
Lord Neuberger suggests that the submissions the QC is making will be clearer when he and his colleagues are able to refer to the transcripts.
That is a good moment to remind people that the transcript of this morning's session will be available at 1600 GMT this afternoon.
Hilary Benn, chairman of the Brexit select committee and Labour MP, told BBC News the 18 month timetable mentioned by the EU's chief Brexit negotiator Michael Barnier meant there was an even more pressing need for transitional arrangements.
Quote MessageThat means there is going to be a very short time from the triggering of Article 50 to negotiate the divorce arrangement and crucially what our new relationship with Europe is going to be once we have left, when it comes to trade and the single market. What I think he has said reinforces the argument that I have been making that we are going to need transitional arrangements because it is very hard to see how you can undo 43 years of relationship and establish a new relationship and do that within just under 18 months."
Image source, Supreme CourtAttorney General Jeremy Wright listens to his colleague Lord Keen
The Scottish government, who we will hear from later, is arguing that it should be, at the very least, consulted before the UK government begins Brexit talks.
But Lord Keen, the UK's chief legal adviser with regard to Scotland, says it is plain from the legislation setting up the Scottish Parliament it has no authority when it comes to matters of international relations, such as EU membership.
As a result, he suggests devolution arguments cannot be used to "qualify or abrogate" the UK government's right to exercise powers to trigger Article 50.
He says: "I would submit these reservations are fatal to reliance on devolution legislation as giving rise to any necessary implication, or indeed any other indication, that the government cannot exercise its foreign affairs and treaty prerogative in the ordinary way."
Pressed by Supreme Court President Lord Neuberger if he is saying "the answer is the same in Scotland as it is here (in the UK as a whole), Lord Keen replies "essentially" it is.
Image source, Supreme Court
Image source, Supreme CourtGovernment lawyer James Eadie has finished for the time being and now we are hearing from Lord Keen, the Advocate General for Scotland, who advises the UK government on legal matters relating to Scotland.
He begins by saying that the devolved administrations, including the Scottish and Welsh governments, have "no competence" in foreign relations and these powers are strictly reserved to the UK government.
This relates specifically, he adds, to the UK's membership of the EU.
Image source, Supreme CourtIn his closing arguments, government lawyer James Eadie says that the "apparent simplicity" of the arguments put forward by those challenging the government's case actually represent what he says is a "constitutional trap".
If it found with his opponents in putting new limits on the government's use of prerogative powers, Mr Eadie suggests the Supreme Court would be crossing a line between "interpretation and judicial legislation".
Quote MessageThe courts would be imposing, in effect, a new control of the most serious kind in a most controversial and, by Parliament, a most carefully considered area."
Image source, Supreme CourtA key moment as government lawyer James Eadie takes head-on the claim that the government would be violating parliamentary sovereignty if it began the process of leaving the EU without Parliament's explicit authorisation.
Quote MessageIt is said the government giving Article 50 notice is an affront to parliamentary sovereignty because Parliament has created rights and only Parliament can alter them. My submission is our case fully respects and offers no affront to parliamentary sovereignty."
In very polite and measured terms, government lawyer James Eadie summarises why he believes the High Court was mistaken in its judgement last month about Parliament's right to invoke Article 50, suggesting its argument was "not sound".
Quote MessageThe court did not take a long-established constitutional principle and apply its inevitable logic. What they did was take a number of different and generally expressed principles and invented a new principle. They took those general principles and, if you will, pressed them into service as absolutes and outside the contexts they were deployed."
Tom Bateman
Political Correspondent, Millbank
Away from the court there's been some fairly big Brexit developments with the EU's chief negotiator Michel Barnier saying that there are only 18 months for negotiations from the point Article 50 is triggered.
This was because, he said, there would need to be a few months at the end for the European Parliament and UK Parliament to agree to whatever deal has been struck.
He also warned that there could be no cherry picking by the UK.
Responding to the comments by Mr Barnier, the Prime Minister's official spokesman said the government had been clear on the process of leaving the EU.
He said: "We will trigger Article 50 no later than the end of March next year. There is then a two year timeframe on that process. We have been clear that we are not seeking to extend that process.
"In terms of how long the negotiations actually take place, clearly that is a matter that will resolve itself as a result of the negotiations."
He said that the position of the EU 27 on the timetable was a "matter for them".
"What's important is that we get the best deal for British business in terms of access to and working within the European markets."
Asked about Mr Barnier's claim that there could be no "cherry picking" of the most favourable parts of EU membership, the spokesman said: "He's not saying anything that hasn't been said before by the European Union."
Asked whether anyone in government was aware that the 27 considered that there were only 18 months for the negotiations he said: "It is the first I have heard of it."
He refused to be drawn on whether the timetable meant the government would not be seeking a transitional arrangement, reiterating that Britain was seeking the "best deal" for businesses.
Government lawyer James Eadie has been on his feet for 90 minutes but gets to sit down briefly while the 11 justices read a note about the UK's departure from the European Free Trade Association (EFTA) to join the EU.
After this is concluded, Mr Eadie says time limitations mean that he would like to pass over the section on "double taxation" and remove it from his submission.
He suggests that the subject is fiendishly complex and that the justices would expect him to "walk them through it and ask him all kinds of difficult questions".
He jokes the decision to strike the issue from his submission reflects his own "cowardice" as well as time constraints, something which seems to find sympathy from members of the Supreme Court as they do not demur.
Government lawyer James Eadie always seems amused when the issue of the Great Repeal Bill gets raised by one of the justices.
Pressed by Lord Sumption, he acknowledges that the process of absorbing all the rights conferred by the 1972 European Communities Act into domestic law before deciding which to keep and which to drop, will be highly complex.
He suggests that there was a degree of "eternal optimism" from the government when it made a statement about the bill in the Commons earlier this year followed by a "pause" about how it may all work in practice.
Asked by Council president Lord Neuberger if there could be "years of excitement" ahead - a comment which brings a ripple of laughter in the courtroom - the QC replies.
Quote MessageThere may be real complexities... and years of real entertainment to come."
An interesting exchange about whether the UK's relationship with the EU, by virtue of the passage of the 1972 European Communities Act, is a "joint effort" between Parliament and the executive.
Lord Wilson cites the case of the wartime Nazi collaborator William Joyce, more commonly known by his nickname Lord Haw-Haw. He tells the court that although the UK declared war on Germany in 1939 through the royal prerogative, Lord Haw-Haw was prosecuted and convicted under the 1352 Treason Act.
Isn't that an example of "joint effort" working in practice?
Government lawyer James Eadie replies: "It was a joint effort in that sense...I accept that there are limitations on lots of these analogies..but what they do illustrate is you need care, care, care before jumping too readily on a big, broad principle however superficially attractive it may seem which says 'you cannot alter the law, you can affect the law'.
"If anything this debate illustrates is that the context needs to be taken into account in these arguments."
Image source, Supreme CourtGovernment lawyer James Eadie is hammering home his central point about the use of prerogative powers.
The broad principle, he says, that the UK government has the executive power to act in international affairs and that Parliament occasionally "takes bites out" of its powers is accepted and backed up by court judgements.
Parliament acknowledges the "suite of powers" conferred on the government while, from time to time, imposing controls on their "ingredients", he adds.