Trump picked only up a single delegate in Wyoming county conventions on April 9 while rival Ted Cruz got nine. In the weekend's state convention, Cruz won all the 14 Republican National Convention delegates up for grabs. Florida Senator Marco Rubio has one delegate and the other four are uncommitted.
"If you don't want to see Donald Trump as the nominee, if you don't want to hand the general (election) to Hillary Clinton, which is what a Trump nomination does, then I ask you to please support the men and women on this slate," Cruz said in his victory speech, holding up a piece of paper of 14 recommended delegates.
For Cruz, the win in Wyoming is another signal that demonstrates how his campaign has organised party insiders and activists to make it difficult for Trump to capture the 1,237 delegates needed to clinch the Republican Party nomination.
With Saturday's sweep, Cruz can count on at least 24 of the 29 delegates from the state. The delegates were chosen by party members rather than ordinary voters.
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However, the real estate billionaire could fall short of the number of delegates needed to secure the Republican nomination for the November 8 presidential election.
That would mean a contested convention where voting for candidates starts again from scratch.
Trump is concentrating on New York, which holds a key primary on April 19.
New York will award 95 Republican delegates while the two Democratic candidates are fighting over 247 delegates in the city.
A number of senior Republican leaders have backed Cruz, a Conservative Texas senator, fearing that Trump's controversial comments make him a weak candidate in the November election.
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Trump hopes to produce a convention that helps to
alleviate questions about his fitness to be president among many Americans but he starts facing an enormous deficit on that issue. The Post-ABC poll found that nearly six in 10 registered voters say he is not qualified to serve as president - with 49 per cent saying they strongly believe that. Meanwhile, Clinton is seen as qualified to serve as president by a 56 per cent majority of voters.
Currently, the Democrats are slightly more united behind Clinton than are Republicans behind Trump. One goal of the Trump campaign is to leave Cleveland at the end of the week with the party more united and enthusiastic about their nominee.
Of seven issues tested, Clinton has double-digit advantages over Trump on three - race relations, handling an international crisis and immigration. Clinton has smaller edges on looking out for the middle class and handling terrorism, while Trump holds small edges on taxes and the economy.
Clinton has a similar edge on empathy with people's problems and representing people's values, and holds double- digit edges on having better judgment and having presidential personality and temperament.
The Post-ABC poll was conducted July 11-14 among a random national sample of 1,003 adults reached on cellular and landline phones. Overall results have a margin of sampling error of plus or minus 3.5 percentage points; the error margin is 4 points among the sample of 816 registered voters.