Missing Madeleine McCann's parents have confirmed they are fighting a Portuguese court decision to side with former police chief Goncalo Amaral over his hurtful claims about her disappearance.

The country's Supreme Court last month rejected their last-ditch appeal over his 2008 book 'The Truth of the Lie' in which he alleged Maddie died in their holiday flat and they faked her abduction to cover up the tragedy.

Judges backed a lower court's April 2016 decision to reverse their 2015 libel win against the ex-detective, leaving the couple facing a huge legal bill and the nightmare prospect of being sued by Amaral.

Kate and Gerry, both 48, have said they will sue if 'The Truth of the Lie' is sold in Britain. Portugal's Supreme Court last month rejected their last-ditch appeal over former detective Goncalo Amaral's 2008 book

And they also challenged Gerry and Kate's insistence they had nothing to with their daughter's disappearance in a devastating put-down which is said to have sparked their fresh legal challenge.

The McCanns have now confirmed they are seeking to get the Supreme Court decision invalidated after launching a formal complaint against the judges' findings.

It was known they had 10 days to file an objection with court officials.

It is thought the McCanns' attempt to nullify the decision is based on comments made by the judges in their 76-page ruling that the 2008 shelving of the Portuguese probe into their daughter's disappearance 'was determined by the fact that public prosecutors hadn't managed to obtain sufficient evidence of the crimes by the appellants.'

The McCanns had their status as 'arguidos' or official suspects lifted on the same day - July 21 2008 - just three days before Amaral published his controversial book.

Judges backed a lower court's April 2016 decision to reverse their 2015 libel win against the ex-detective (pictured left). He alleged Maddie (right) died in their holiday flat and her parents faked her abduction to cover up the tragedy

Correio da Manha reported: 'The McCanns have requested the annulment of the Supreme Court decision, terming it frivolous for saying it 'had not been possible for public prosecutors to obtain sufficient evidence of crimes by the appellants.''

The newspaper said the McCanns had described the ruling as 'leviano' in the complaint lodged through their Portuguese lawyer - which in English translates as 'frivolous' but can also mean 'sloppy' or 'rash'.

No-one from the Supreme Court was available for comment this morning.

The McCanns' lawyer Isabel Duarte, asked if she had lodged a formal complaint against the Supreme Court ruling, confirmed: 'We delivered it.'

It was unclear today if another set of Supreme Court judges dealt with complaints about rulings - or they were handed to another judicial body to deal with.

Amaral was ordered to pay the McCanns €500,000 euros (£430,000) by a Lisbon court in April 2015 after they won round one of their lengthy judicial battle over his book and a subsequent TV documentary.

The McCanns' lawyer Isabel Duarte (pictured), asked if she had lodged a formal complaint against the Supreme Court ruling, confirmed: 'We delivered it'

The former police chief got that ruling - and a ban on selling his book - overturned on appeal in April last year.

The decision by Lisbon's Court of Appeal sparked the Supreme Court fight which was resolved on January 31.

The full 76-page ruling said to have sparked a new legal challenge by the McCanns was released just under a week later.

Judges made it clear in their decision their job was not to decide whether the McCanns bore any criminal responsibility over their daughter's disappearance and it would be wrong for anyone to draw any inferences about the couple's guilt or innocence from their ruling.

But they added: 'It should not be said that the appellants were cleared via the ruling announcing the archiving of the criminal case.

'In truth, that ruling was not made in virtue of Portugal's Public Prosecution Service having acquired the conviction that the appellants hadn't committed a crime.

'The archiving of the case was determined by the fact that public prosecutors hadn't managed to obtain sufficient evidence of the practice of crimes by the appellants.

'There is therefore a significant, and not merely a semantic difference, between the legally admissible foundations of the archive ruling.

'It doesn't therefore seem acceptable that the ruling, based on the insufficiency of evidence, should be equated to proof of innocence.'

They added, highlighting the McCanns' Tapas Nine friend Jane Tanner's much-questioned sighting of the suspected 'abductor': 'It's true that the aforementioned criminal inquiry ended up being archived, namely because none of the apparent evidence that led to the appellants being made 'arguidos' was subsequently confirmed or consolidated.

'However even the archive ruling raises serious concerns relating to the truth of the allegation that Madeleine was kidnapped.'

The Supreme Court judges said the McCanns claimed Amaral's book and the TV documentary based on the book formed no part of case files made public in 2008 and would have damaged the honour and good name of any 'innocent person who had been cleared through the shelving of the criminal investigation.'

But they stated: 'We consider the invocation of the violation of the principle of innocence should not be taken into account here, since this issue is not relevant to the resolution of the question that needs to be decided here.'

The Supreme Court judges said the McCanns claimed Amaral's book and the TV documentary based on the book formed no part of case files made public in 2008 and would have damaged the honour and good name of any 'innocent person'

They said the 'crucial question' for them was how to resolve the rights of Kate and Gerry McCann to their 'good name and reputation' and the rights of Goncalo Amaral and the other respondents including the book editors to the constitutionally-inshrined right of 'freedom of expression.'

Concluding Amaral had not acted 'illicitly,' they ruled his book was not a personal and unjustified attack on the McCanns with a 'defamatory intention' behind it which would not be protected by freedom of speech rights.

Describing the book and the TV documentary based on it as an 'opinion' based on the logic of facts and evidence contained in the criminal case files, they added: 'Our opinion is that rather than an injurious animus, the intention was informative and defensive.'

The Supreme Court ruling meant Amaral was spared having to pay the McCanns the compensation he was ordered to hand them after the first court ruling in 2015.

Missing Madeleine McCann's parents are fighting a Portuguese court decision

The payment was frozen when he launched his successful appeal.

Earlier this month it emerged the ex detective, removed as head of the investigation into Madeleine's May 3 2007 disappearance after criticising British detectives, was writing a new book about the unsolved mystery.

It is understood he will be critical in the new book of some of the things Scotland Yard did in their review and later ongoing investigation of the case.

The former cop insisted from day one of his court fight with the McCanns that everything he wrote in his book was based on the publicly-available case files.

Kate and Gerry, both 48, of Rothley, Leicestershire, have said they will sue if 'The Truth of the Lie' is sold in Britain.

They said in a statement after learning of the Supreme Court ruling against them: 'What we have been told by our lawyers is obviously extremely disappointing.

'It is eight years since we brought the action, and in that time the landscape has changed dramatically, namely there is now a joint Metropolitan Police and Policia Judiciaria investigation which is what we have always wanted.

'The police in both countries continue to work on the basis that there is no evidence Madeleine has come to physical harm.

'We will of course be discussing the implications of the Supreme Court ruling with our lawyers in due course.'

It is believed the McCanns are discussing the possibility of taking the case to the European Court of Human Rights.

The 20,000 page 'Madeleine files made public in 2008 contained a report by public prosecutors which said: ' No element of proof whatsoever was found which allows us to form any lucid, sensible, serious and honest conclusion about the circumstance of Madeleine's disappearance from the apartment...including, and most dramatically, establishing whether she is alive or dead, which seems more probable.'

Madeleine McCann disappeared from a holiday apartment in Praia da Luz, a resort in the Algarve region of Portugal, on May 3, 2007

Referring to the McCanns' much-criticised decision to leave their daughter, then three, alone with younger siblings Sean and Amelie while they ate tapas nearby, it added: 'We must also recognise that the parents are paying a heavy penalty over the disappearance of Madeleine for their carelessness in monitoring and protecting their children.'

Portuguese police chiefs said late last year they were 'completely in tune' with British detectives still investigating Madeleine's disappearance, appearing to end years of tension between the two forces whose theories on the youngster's fate have differed wildly.

Portuguese prosecutors reopened their probe into Madeleine McCann's disappearance in May 2014, and are now working in close coordination with Scotland Yard's scaled-down Operation Grange probe into Madeleine's fate.