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Cross's (Vecten's) theorem

First try and explain why (prove) it is true yourself. But if you get stuck, have a look at the (unedited) proof of one of my Kennesaw graduate students at *Gin Chou's proof*

Then explore the analogous cases below and also try to explain why (prove) why they are true. (Each one below, except the last, has a link to an article of mine on the topic).