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19 Junio 2017, 03:06 | Crisanna Felipe
Under the proposed deal, the DUP would likely support May's Conservatives on big issues such as the budget, Brexit and defence legislation on a vote-by-vote-basis.
The deal May has brokered with the DUP leader Arlene Foster involves giving an economic boost to Northern Ireland, but agreeing to disagree on some social policy issues, including LGBT rights and same sex marriage.
"We are continuing to have talks but today, as you will imagine, there has been a real focus on this awful tragedy in London", May said.
Meanwhile in London, the former prime minister Sir John Major joined a growing chorus of voices telling Theresa May she does not have a mandate to pursue the hard Brexit she was planning.
Cameron resigned from his post after the defeat in the Brexit as 52% of people voted for an exit from the European Union (EU).
The Irish republican party visited London to hold meetings with ministers and to set up their parliamentary office but the party maintained its traditional refusal to take its seats out of opposition to Westminster's jurisdiction in Northern Ireland and the oath all MPs must make to the Queen.
Prime MinisterTheresa May has made a joke at her own expense as Britain's House of Commons got underway - a reflection of her new humility following a disastrous snap election in which she lost her majority. Ms Foster said she hoped a deal could be done "sooner rather than later".
Referring to speculation that negotiations could be extended until March 2019, Barnier said: "If we work seriously, I see neither the usefulness nor the interest of pushing back this date".
During the election campaign, May had used the "coalition of chaos" phrase to describe what a victory for Labour would look like.
May told the first post-election session of the lower house that she had discussed the DUP talks with her cabinet.
"Now a third of our parliamentary party is female and we have our most diverse group of MPs ever, I feel our MPs form a more representative group to elect a deputy leader". He also said Brexit talks could possibly be delayed.
Mrs Foster was nearly certain to ask for greater investment in Northern Ireland as part of the deal, though changes to security and legacy issues from the Troubles were unlikely to be included in a pact.
"Bringing stability to the United Kingdom government in and around issues around Brexit, obviously around counter-terrorism, and then doing what's right for Northern Ireland in respect of economic matters".
Even the idea of an alliance is complicated, however.
But a deal with the DUP also risks destabilising Northern Ireland by increasing the influence of pro-British unionists.
Foster's rivals in Northern Ireland, such as Sinn Fein's Gerry Adams, have objected.
"I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements", Ms O'Neill said in a statement.
The stakes for May are high.
Yet many of her lawmakers and party members favour a sharp break with the European Union - a sign of the divisions over Europe that helped sink the premierships of May's predecessors Margaret Thatcher, John Major and Cameron.
Labour's unexpectedly strong second-place showing has thrown national politics into disarray.
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