I'm a little confused by the axioms.
Surely the "unique" folds in the first few examples should allow for the mirror-image fold, both mountain and valley bringing the points together?
And Axiom 5 doesn't even work if P1 lies in a circle touching L1 and centred on P2!
Axioms 6 and 7 don't hold if L1 and L2 are parallel.
So, not exactly Axioms, then.