Mr Paisley was speaking at the opening of a new ferry terminal
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First Minister Ian Paisley has said he believes Peter Robinson will become first minister on Thursday.
He said it would be an "evil thing" if anyone "so late in the day would now say no, we're stopping it".
Mr Paisley was speaking as he arrived to open a new multi-million pound ferry terminal in Belfast alongside Martin McGuinness.
There has been speculation Sinn Féin may not renominate Mr McGuinness when Mr Paisley stands down.
The party is widely acknowledged as being frustrated that the DUP is using its veto to block a range of issues, including the devolution of policing and justice.
Mr McGuinness said he hoped the outstanding issues could be resolved before Thursday.
"What we are dealing with is the need for all sides in this equation to recognise, 18 months on from the St Andrews Agreement, the absolute urgency required to see that agreement fulfilled," he said.
"Discussions are continuing as we speak. We will continue for as long as it is necessary to see the situation in a fashion that will convince people that there is an urgent desire to have the St Andrews Agreement fulfilled."
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Both have too much to lose, particularly the DUP in terms of having to face an electorate who are unhappy with what they did
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If Sinn Féin refused to nominate a deputy first minister, Peter Robinson would not be able to become first minister.
The two parties would then have seven days to resolve the issue and if they remained deadlocked, the secretary of state would have to step in and call an election to resolve the impasse.
Prime Minister Gordon Brown said he had had talks with the leaders of "all the parties who are in the administration in Northern Ireland".
"I hope we can move forward tomorrow (Thursday) and that the new first minister will be nominated as will the deputy first minister. I believe that this can and will happen."
Jim Allister, leader of Traditional Unionist Voice, said he believed that both Sinn Fein and the DUP have too much vested in the present executive and would want to avoid an election.
"Both have too much to lose, particularly the DUP in terms of having to face an electorate who are unhappy with what they did," he said.
Stena Line's new Belfast terminal is the most expensive engineering project in the port's history.
Its official opening is also being attended by Mr Paisley's Scottish counterpart Alex Salmond.
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