Mrs May said in her letter: "To put it frankly, if we are to make good progress in the negotiations and deliver the smooth, orderly and successful Brexit the country wants, we need to agree with the European Union the terms of the backstop".
The note said: "The UK is clear that the temporary customs arrangement, should it be needed, should be time limited, and that it will be only in place until the future customs arrangement can be introduced".
David Davis has shown how keen he is to get shot of the whole thing, the prime minister has only narrowly avoided a complete meltdown in her government and the country has been told via a white paper that its prosperity is being sacrificed under the altar of Brexit extremism.
Underlying the debate is the goal of ensuring that the border between Northern Ireland and the Republic of Ireland remains open, with no border posts or customs checks, after Brexit to protect economic links and peace on the island.
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It would be able to implement only "those parts of new free trade deals not covered by this temporary customs arrangement". We will examine it with 3 questions: is it a workable solution to avoid a hard border?
Writing for Express.co.uk on May 31, leading Brexiteer Jacob Rees-Mogg echoed the notion that the United Kingdom needs to fully leave the customs union to honour the Brexit vote.
The proposal was published after a series of crunch meetings between Brexit Secretary David Davis and Prime Minister Theresa May, with Davis reportedly angry that the initial draft did not place a time limit on the length of the "temporary" backstop.
The idea of the backstop is controversial with eurosceptics, who fear it will keep Britain tied to the EU's rules for years after it leaves the bloc on March 29, 2019. Is it an all-weather backstop?
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"The backstop paper has been amended and now expresses, in much more detail, the time limited nature of our proposal - something the PM and DD have always been committed to".
Labour said that among the amendments was one that removed the requirement for the government to hold a "meaningful vote" in parliament on the deal it will try to negotiate with the European Union for Britain's future relationship with the bloc.
The Prime Minister also held separate face-to-face discussions with leading Brexiteers Boris Johnson and Liam Fox.
Asked if she expected Davis to remain in his position on Thursday, May's spokeswoman said: "Yes... of course".
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"The truth is that leaving the EU's customs union is necessary to keep faith with the referendum result and reap the benefits of Brexit".