EU leaders voiced their respect for British prime minister Theresa May after the Conservative leader on Friday (24 May) announced that she would step down on 7 June.

She will remain as prime minister while her Conservative party undertake a new leadership contest, which is expected to take around six weeks.

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May's doorstep announcement from Downing Street on Friday morning also raised concerns that her departure increases the likelihood of the UK leaving the EU without a divorce deal when the extension to October 31 runs out.

The EU also made it clear that a change in leadership does not change Brussels' position on the withdrawal agreement on offer, which was negotiated with May for over a year and has been rejected now three times by the UK parliament.

"The [EU commission] president [Jean-Claude Juncker] very much liked and appreciated working with prime minister Theresa May. She is a woman of courage, for whom he has great respect. He will equally respect and establish working relations with any new PM, whomever they may be," a commission spokesperson said in Brussels.

German chancellor Angela Merkel noted May's decision "with respect", a German government spokesperson said Friday.

"Germany continues to wish for an orderly British exit from the EU. This would require a successful vote in the House of Commons," the spokesperson added.

The EU in April granted an extension to the Brexit deadline for the UK until the end of October. If the UK fails to pass the withdrawal agreement by then, it could leave the EU without any deal setting out the terms of the divorce - making the next phase of negotiation on the future relationship very difficult, and likely provoking a severe economic shockwave in the UK.

"The working assumption is still that Brexit happens on the 31st of October, as the willingness has been expressed by the voters in the UK," a commission spokeswoman added.

The EU has already said that the three key issues in the withdrawal agreement - a financial settlement, citizens' rights and the Irish border - aren't going away, even if Britain leaves without a deal, and will have to be resolved as part of the new round of talks on the future.

Macron prod

French president Emmanuel Macron in a statement urged "rapid clarification" from Britain on what it wants from Brexit. He added that he stands ready to work with a new British prime minister, but that "the principles of the EU will continue to apply, with the priority on the smooth functioning of the EU."

Dutch prime minister Mark Rutte said that the "the deal reached between the European Union and Britain for an orderly Brexit remains on the table."

The EU's chief negotiator, Michel Barnier, also said he had "full respect" for May and her determination "in working towards the UK's orderly withdrawal".

Despite the words of respect and well wishes on Friday, EU officials and leaders have been growing frustrated with May over the course of Brexit negotiations.

Her move to first launch the Brexit countdown clock by kickstarting the Article 50 procedure, without first deciding the UK negotiating position, and then call a general election - which wanted several months of crucial time for negotiations and produced a loss of a working majority for May - was met with surprise.

May's threat to use security arrangements with Europe as bargaining chip in the negotiation was met with dismay, as well as her insisting on trying to negotiate with individual member states by going behind the commission's back - which ended in failure.

EU leaders have asked her repeatedly to clarify what Brexit means, what does the UK want - without receiving concrete or consistent answers.

The EU gradually lost trust in her ability to forge a political consensus in the UK over Brexit and the divorce deal, which was highlighted most painfully, when at the end of 2017 she delayed her agreement to the outline of what later became the Brexit deal because of the insistence of her Northern Irish unionist allies.

EU officials and politicians - in whose DNA consensus-building in encrypted - watched with resignation, as May failed to force a consensus at home, even within her own party, let alone with the Labour opposition or the half of the UK which voted Remain, behind her Brexit approach.

On Friday, however the Irish prime minister Leo Varadkar already sounded the alarm that what and who might come next, could be worse, and make the next phase of talks "very difficult" for Ireland.

"We may see the election of a eurosceptic prime minister who wants to repudiate the withdrawal agreement and go for no deal or we may even see a new British government that wants to see a closer relationship with the EU and goes for a second referendum," Varadkar said.

"Politicians throughout the EU have admired her tenacity, her courage, and her determination during what has been a difficult and challenging time," Varadkar added on May.

Spain also warned that a no-deal scenario seems almost impossible to avoid now.

"The British government, the British parliament" would be "solely responsible for a no-deal exit and its consequences," a Spanish government spokeswoman said.

Although not as serious as the Northern Irish border issue, a centuries-old row between the UK and Spain over the rock of Gibraltar is one of the many difficult issues sparked by the UK's decision to leave the bloc.