The European Union's chief Brexit negotiator has promised to deal with negotiations on the UK's withdrawal in a "constructive" way and said he believed a "fair" deal was possible for both sides and "far better" than the prospect of the UK leaving without an agreement.
Michel Barnier was speaking at the end of the first day of negotiations in Brussels with Brexit Secretary David Davis, who said he was "optimistic" of reaching a good deal.
Mr Davis said Prime Minister Theresa May would brief fellow EU leaders at a summit on Thursday on the UK's approach to the rights of expatriate citizens, which will be set out in detail in a paper on Monday.
Mr Barnier made clear the talks, beginning a year after the UK's Brexit referendum, would take place according to a timetable set by the EU, under which progress on the terms of withdrawal must be made before any discussions on a future trading relationship.
Speaking alongside Mr Davis at a Brussels press conference, he said: "For both the European Union and the United Kingdom, a fair deal is possible and far better than no deal.
"That is what I said to David today.
"That's why we will work all the time with the UK and never against the UK. There will be no hostility on my side.
"I will display a constructive attitude firmly based on the interests and support of the 27."
The first stage of negotiations will cover the issue of the rights of EU citizens in the UK and Britons living in other EU countries, as well as the "single financial settlement" of outstanding liabilities and issues surrounding the border between Northern Ireland and the Republic, he said.
Mr Barnier did not put a figure on the settlement likely to be required from the UK, estimated by some in Brussels at as much as 100 billion euro (£88bn).
But he made clear that only when the other 27 member states are satisfied that sufficient progress is being made on this issue that the talks can move on to the future trade relationship.
He said: "We have to commit ourselves now mutually to guarantee rights to citizens on either side of the Channel so they can continue their lives as in the past.
"We have to clear the accounts and we have to honour our mutual financial commitments.
"We also have to find solutions to maintaining all the commitments of the Good Friday Agreement.
"It is by lifting uncertainties around these issues that we will lay the foundation and create the climate of trust which will enable us to build a new partnership."
Discussing his hopes for the talks, Mr Barnier quoted one of the European Union's founding fathers, Jean Monnet: "I'm neither optimistic nor pessimistic, I'm determined."
And in reply, Mr Davis quoted Sir Winston Churchill's dictum that "the pessimist sees difficulty in every opportunity. The optimist sees the opportunity in every difficulty".
"Bridging between Churchill and Monet, I am certainly a determined optimist," said Mr Davis.
He added: "Today marks the start of a journey, for the United Kingdom and for the European Union.
"There is a long way to go, but we are off to a promising start. We have taken the first, critical steps together."
"Now, we have a shared responsibility to deliver quick and substantive progress."
Mr Davis characterised discussions in Brussels on Monday as "very productive".
"We have laid a solid foundation for future discussions, with an ambitious but eminently achievable timetable," said the Brexit Secretary.
"It was clear from the opening that both of us want to achieve the best possible outcome and the strongest possible partnership. One that works for the UK and for the EU.
"And we agreed that we stand a much greater chance of success if our teams work together as that's been demonstrated today."
Mr Barnier and Mr Davis will meet every four weeks over the coming months, bringing their teams together for a matter of days each time.
Different working groups will negotiate on the issues of expat rights, the financial settlement and the Irish border.
Mr Davis said talks over the Irish border had taken more time than anything else on Monday, with discussions on the "political sensitivities" of the issue and the "determination to maintain as near as possible an invisible border".
Answering questions, the Brexit Secretary defended the decision to make the issue the subject of a dialogue rather than a working group.
"This is a technically difficult issue, but it's one which I am certain is soluble or that will probably take us until the end of the process when we have already decided what our customs and free trade arrangements are," he said.
"But that's why we're starting now and that's why this needs to be dealt with by dialogue of the two highest ranking people, the two coordinators in this team.
"There's no doubt in my mind that this is right at the top of the priorities we need to resolve in this negotiation."
Mr Davis also denied suggestions the agreed timetable showed Britain's "weakness" and insisted it is "completely consistent" with the Government's aim of parallel trade and exit talks.
"It's not when it starts it's how it finishes that matters," he said.
"The UK has been crystal clear in our approach to the negotiations, the withdrawal process cannot be concluded without the future relationship also being taken into account.
"They should be agreed alongside each other, this is completely consistent with the Council's guidelines which state nothing is agreed until everything is agreed."
Mr Davis also brushed off the idea Britain's negotiating stance could change given political instability in the UK, where Mrs May is seeking support of the Democratic Unionist Party (DUP) to prop up her minority government.
The Brexit Secretary said: "The position hasn't changed, we have the Lancaster House speech, the two white papers, and the Article 50 letter, all backed up by a manifesto too.
"So it's the same as it was before."
Mr Barnier agreed and said Mr Davis had made an "important point".
Speaking partly in English and partly French, Mr Barnier said it was useful for him at Monday's meeting to be able to "start off on the right foot, as the clock is ticking".
He and Mr Davis had agreed the common objective of agreeing on the principles of the key challenges in the first phase of discussions "as soon as possible", he said.
"The European Council under the chairmanship of President (Donald) Tusk can then decide on whether we can show sufficient progress or not and we can move to scoping the future relationship on trade and other matters," he said.
Mr Barnier said: "We must lift the uncertainty caused by Brexit.
"We want to make sure that the withdrawal of the UK happens in an orderly manner."
He added: "In leaving the European Union, as the United Kingdom has done in exercising its sovereignty, the UK will no longer enjoy the same rights and advantages as EU member states.
"I believe though that it is very much in our joint interests to build a new partnership between the 27 and the UK, and I am convinced that this new partnership will contribute to stability on our continent.
"We, the 27 and the European institutions, are united in our negotiating stance and in our efforts to achieve this goal."
Terms of reference agreed by both sides envisage four rounds of talks on the first phase of discussions, in the weeks starting July 17, August 28, September 18 and October 9, implying trade talks are unlikely to open until after the European Council summit of October 18/19.
Both English and French will be used as working languages in negotiations and working documents, and the two sides have agreed a set of principles on the openness of their talks.
The terms of reference state "for both parties the default is transparency" and that it is for the side providing any information to state whether it should remain secret.
"Where possible, both parties will seek to agree public statements relating to negotiating rounds," the document stated.
Asked if he had made any concessions to Britain, Mr Barnier said: "I am not in a frame of mind to make concessions, or ask for concessions.
"It's not about punishment, it is not about revenge.
"Basically, we are implementing the decision taken by the United Kingdom to leave the European Union, and unravel 43 years of patiently-built relations.
"I will do all I can to put emotion to one side and stick to the facts, the figures, and the legal basis, and work with the United Kingdom to find an agreement in that frame of mind."
He added: "The United Kingdom has decided to leave the European Union, it is not the other way around.
"The United Kingdom is going to leave the European Union, single market and the customs union, not the other way around.
"So, we each have to assume our responsibility and the consequences of our decisions.
"And the consequences are substantial."
Liberal Democrat leader Tim Farron said: "David Davis said the row of the summer would be over the sequencing of Brexit talks, and one day in he has capitulated.
"The man is a joker.
"Despite the Government's posturing, the EU was clear today it has not made a single concession to David Davis. He has been utterly humiliated.
"After Philip Hammond tried to rugby tackle his Conservative colleagues before they leap off the cliff edge on trade, David Davis has shrugged him off and dived into the abyss.
"He has announced his intention to ruinously rip us out of the single market and customs union, despite the Government's own Chancellor warning this would be very, very bad for Britain."