Outlining the five main principles of her "fair and serious offer", May told other European Union leaders at a summit in Brussels that she wanted to offer certainty to European Union citizens about their future in Britain, again using a softer tone in her approach to Brexit.
UK Brexit Secretary David Davis said that the Brexit talks had gotten off to a "promising start" even though it has become obvious that Britain conceded to the EU's preferred order.
European Union President Donald Tusk echoed rocker John Lennon's "Imagine" Thursday as he said he hoped Britain might still stay in the bloc.
"A big hurdle, a big step has been made in the first week of Brexit negotiations - it's really now for the European Union to respond", he said. That is now the case.
The EU commission will let the 27 member states know what exactly has happened on Brexit a week after the negotiations began.
"Our optimism should still be extremely cautious but we have good reason to talk about it", Tusk said earlier.
"There is a long way to go, but we are off to a promising start. They will be pursued in a normal way".
He stressed that although the United Kingdom has strong objectives, so does Ireland, and efforts will be made to avoid "an economic border".
Germany's Europe minister, Michael Roth, said in Luxembourg that the opening of the negotiations had shown British leaders still needed a "reality check" on what they could achieve.
"It's a pity that this decision was made but we can not turn and look only backwards". SOFTER TONE Weakened by an election she did not need to call, May has watered down her government's program to try to get it through parliament and set a softer tone in her approach to Brexit.
Earlier, finance minister Philip Hammond sought to soften the more confrontational tone towards the European Union taken by May, saying Britain needed to put business and jobs first, and agree a transitional arrangement to avoid an economically damaging cliff-edge exit.
Central Portugal forest fires: 61 Dead, PM Calls Disaster a 'Tragedy'
Spain and France are sending aircraft to assist in efforts to douse the flames, authorities say. "The EU is fully ready to help". A burnt truck next to a gate in the village of Pobrais, near Pedrogao Grande , central Portugal , Monday, June 19 2017.
Britain's political instability has caused concerns in European capitals hoping to get the negotiations over with quickly and cleanly, as has May's oft-repeated threat to walk away from the talks without a deal if necessary.
"The doors are still open. If they want, if the government wants to, when they see now all the consequences of Brexit".
"I am not a dreamer. So we are waiting now". "President Dalia Grybauskaite of Lithuania, which has over 100,000 citizens in Britain, insisted relations would remain close and tweeted the Motown lyric: "#Brexit: "ain't no mountain high enough".
"I am not a dreamer and I'm not the only one. We can speculate, but it is a waste of time".
"There will be no hostility on my side, I will display a constructive attitude firmly based on the interests and support of the 27".
"But it all depends of course on what Theresa May and her team will decide".
"Yeah, we all have that dream".
"I hate Brexit from every angle".
Also on the agenda in Brussels will be counter-terrorism co-operation, sanctions on Russian Federation and Europe's response to the USA withdrawal from the Paris Accord on climate change.
"The control of certain foreign direct investments in sensitive areas is mentioned". The EU proposal is what we have been asking for all along. "That's something I really care about".
Apart from the issue of European values, Macron and the easterners also clash over the question of posted workers, where the French president wants to push for higher wages - which eastern member states oppose vehemently.
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