Theresa May has urged MPs and peers to “remember” the result of the referendum on leaving the EU.
The prime minister said the government was “getting on” with the task, after it was told by the High Court that Parliament must have a vote before the formal Brexit process gets under way.
The businesswoman who brought the case said the ruling meant “certainty”.
But UKIP leader Nigel Farage warned of protests on the streets if the UK’s referendum decision was ignored.
The High Court ruled on Thursday that Parliament should vote on when the government could trigger Article 50 of the Lisbon Treaty – which will begin the formal process of leaving the EU.
Mrs May has promised to get this done by the end of next March.
And ministers have vowed to fight to get the ruling overturned next month in the Supreme Court.
Labour leader Jeremy Corbyn said his party would block the triggering of Article 50 in Parliament if Mrs May did not guarantee access to the single market.
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Speaking on BBC One’s Andrew Marr Show, Gina Miller, the investment manager who brought the High Court case against the government, said: “Everyone in this country should be my biggest fan, because we have used our own money to create certainty about the way ahead.”
She added: “Do we want a country where we have no process?”
But Mr Farage said the court’s decision meant the country was faced with “half Brexit”, adding that the “reach of the European Union into the upper echelons of this country makes it quite difficult for us to trust the judgement”.
He warned: “If the people of this country think that they’re going to be cheated, they’re going to be betrayed, then we will see political anger, the likes of which none of us in our lifetimes have ever witnessed.”
Asked if there was a danger of disturbances in the street, he replied: “Yes, I think that’s right.”
Mr Farage said: “The temperature of this is very, very high. I’m going to say to everyone who was on the Brexit side, ‘Let’s try and get even. Let’s have peaceful protests and let’s make sure, in any form of election, we don’t support people who want to overturn this process.'”
Analysis – Chris Mason, political correspondent
Political reporters aren’t meant to spend much time reporting from courtrooms.
But it’s an illustration of how febrile politics is at the moment that it felt like I spent half the summer in court covering the rows over the Labour leadership race.
And now the decision of the High Court is posing the biggest questions of all for the government.
Could Brexit provoke a constitutional crisis over who’s in charge and how and by when our EU departure will happen?
And does that mean there could be a general election as soon as early next year?
This may yet all blow over quite quickly.
But equally, it might just blow up into the line of mega row that makes a general election unavoidable.
Speaking at Heathrow Airport as she left for a trade mission to India, Mrs May said: “I think we all have to remember, and what MPs and peers have to remember, is that we had a vote on 23 June.
“The British people, the majority of the British people, voted to leave the European Union. The government is now getting on with that.”
She added: “I want to ensure that we get the best possible deal for the UK as we leave the EU, that’s the best possible deal for trading with and operating within the single European market.
“But alongside that, the UK will be a confident, outward-looking nation, taking its place on the world stage, looking to build relationships around the globe.”
However, Mr Corbyn said Labour would block the prime minister from triggering Article 50 unless she agreed to the party’s “Brexit bottom line”, which includes access to the European single market.
He told the Sunday Mirror: “The court has thrown a big spanner in the works by saying Parliament must be consulted. We accept the result of the referendum.
“We are not challenging the referendum. We are not calling for a second referendum. We’re calling for market access for British industry to Europe.”
Mr Corbyn said the opposition would not allow Article 50 to go ahead unless Mrs May agreed four principles:
- access to the single market
- a commitment to EU workplace rights
- guarantees on safeguarding consumers and the environment
- A pledge to commit funds for any EU capital investment lost by Brexit
The Labour leader said his party “would be ready” if the government decided to call an early election.
The judges behind the High Court ruling against the government have faced a backlash from some newspapers which led the Bar Council to urge the government to curb the criticism.
Lord Chancellor Liz Truss backed the independence of the UK’s judiciary but stopped short of condemning the attacks.
Mrs May has said she is “confident” the government will win the appeal and is committed to triggering Article 50 by March 2017.
We must remember Brexit vote, warns Theresa May – BBC News