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We can prove by induction the "computing laws" in N:
forall x in N, y in N, z in N: x+(y+z) = (x+y)+z.We take arbitrary x in N and y in N and prove by induction on z.
forall z in N: x+(y+z) = (x+y)+zWe have to show
x+(y+0) = (x+y)+0....
See lecture notes.