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Natural Number Laws

For all natural numbers x, y, z, we have:

x + 0	=	x,
x * 1	=	x,
x + y	=	y + x,
x * y	=	y * x,
x + (y + z)	=	(x + y) + z,
x * (y * z)	=	(x * y) * z,
x * (y + z)	=	(x * y) + (x * z),
x	<=	x,
(x <= y /\  y <= x)	=>	x = y,
(x <= y /\  y <= z)	=>	x <= z.