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Corbyn's not pro-'Brexit'. He has re-iterated, even after Labour took the position is has (which is shifting) that he is personally in favour of 'remain and reform'. However, there has been a referendum decision (whatever stuff about illegal influencing of that is currently coming out) and they have said they have to respect that.
I still think they've successfully seized the high ground that way, as by getting bogged down in all that utter nonsense they would never have been able to build the policy platform that saw them do reasonably well in the last general election. May, bizarrely, wanted to make that election about '"Brexit" or not "Brexit", and they didn't take the bait. The Lib Dems imagined they might make electoral gains by pledging to 'reverse "Brexit"', and that didn't work for them.
Political issues go on the boil and then they go off the boil if you let them. If you keep up a major controversy, they will not go off the boil.
It's far from an ideal situation, but as I've said before, they've been negotiating a minefield.
Agreed, I'm still not sure why he is so pro-Brexit, it definitely seems to go beyond simply trying to appeal to those in the party/country who voted leave.
However his sentiment in that speach is spot-on: immigration has been good for the UK and using it to deflect away from complicated issues such as housing and employment rights is just a cheap race to the bottom that poisons proper debate.