Yes, but wouldn't there be an induced current which creates extra magnetic field lines around/ within the wire?
Ah okay, I see your point. There was a decrease (purely from the Magnetic field within the wire changing), however Lenz's law should result in an increase, to counteract the change!
The thing to know is that
the field induced by Lenz's law will not make up for the original change. I'm not sure if you do Chemistry, but it's sort of like Le Chatelier's principle if you do. If you change the flux through a loop, the current will try to reverse that change, but not so strongly as to actually reverse it. Sorry, I understand your point now, and I hope you understand my explanation (well, it's less of an explanation, more of a 'this is just how it is')!
Jake