Alfie Evans' parents, who live in Liverpool, are using a piece of ancient English common law during the latest stage of their fight for treatment.
Tom Evans (pictured below wearing a blue coat) , 21, and Kate James, 20, want to move their son from Alder Hey Children's Hospital in Liverpool to a hospital in Rome.
The couple, which is being represented by the Christian Legal Centre (CLC), say Italian doctors are willing to treat the little boy and an air ambulance is available.
They are now arguing that Alfie is being wrongly "detained" at Alder Hey and have made a habeas corpus application.
A writ of habeas corpus - Latin for "you may have the body" - is a legal manoeuvre which requires a court to examine the legality of a detention.
It is a piece of common law which probably dates back to Anglo-Saxon times.
A High Court judge and three Court of Appeal judges have dismissed their claim.
But the couple might make an application to the Supreme Court.
Alfie's parents have already lost fights in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
In February, Mr Justice Hayden ruled that doctors at Alder Hey could stop treating Alfie against the wishes of his parents following hearings in the Family Division of the High Court in London and Liverpool.
Specialists at Alder Hey said life-support treatment should stop and Mr Justice Hayden said he accepted medical evidence which showed that further treatment was futile.
Mr Justice Hayden said flying Alfie to a foreign hospital would be wrong and pointless.
Court of Appeal judges upheld his decisions.
Supreme Court justices and European Court of Human Rights judges refused to intervene.
Last week, Mr Justice Hayden endorsed a detailed plan put forward by Alder Hey doctors for withdrawing life-support treatment - after considering a number of issues at a follow-up High Court hearing.
He also dismissed the suggestion that Alfie was being unlawfully detained an Alder Hey and refused to grant a writ of habeas corpus.
Three Court of Appeal judges on Monday dismissed a challenge to Mr Justice Hayden's decision after analysing argument at a hearing in London.
But barrister Paul Diamond from the CLC, who represented Alfie's parents, said the couple might make an application to the Supreme Court.
Appeal court judges said doctors should continuing treating Alfie pending any Supreme Court decision.
Judges have heard that Alfie, born on May 9 2016, is in a "semi-vegetative state" and has a degenerative neurological condition doctors had not definitively diagnosed.
Specialists say his brain has been "eroded".
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