I support the government's right to appeal. I don't, yet, see any reason to suppose that their appeal will succeed.
PP has already put up the court transcript for the first day. It makes for a lot of reading, and I've only skimmed bits (179 pages or so). First impressions are that the Government's case is the same as it was before, perhaps adding an appeal to populism, eg from the end of today's proceedings:
Mr Eadie (on behalf of HMG): "... If you said to the ordinary man or woman on the street, 'Do you regard the fact that a referendum has occurred to be remotely relevant to the question of whether or not the Government can give Article 50 notice?' the answer would be, 'Of course it is.'"
Lord Neuberger: "If you put it as 'remotely relevant' --"
Mr Eadie: "Relevant as a matter of law."
Lord Neuberger: "As a matter of law, they will probably say, 'I will ask a lawyer.'"
Anyway, there are three more days of submissions to go, and then a month or so of deliberations before the Supreme Court ruling, but if this particular appeal is a "farce", then you should look to HM Government for trying to appeal a decision on the grounds that, "I just don't agree with the last one but only because I'm stubbornly sticking to my already-rejected argument."