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We prove p(x, y) : <=> x+y is even is an equivalence relation on N.
(1) x+y is even /\ y+z is even.We have to show (2) x+z is even. From (1), we have some a in N and b in N such that
(3) 2a = x+y /\ 2b = y+z.Thus we know (2) because of
x+z = (x+y) + (y+z) - 2y = 2a + 2b - 2y = 2(a+b-y).