SET® Mathematics
Proof of Theorem 1.3

(an * cn) (an * bn) = an(cn * bn)  

There exist four options: a = b = c a = b, bc a b, b = c abc  a = b = c (1*1)(1*1) = 1*(1*1) (1*1) = (1*1) 1 = 1 a = b, bc (1*1)(1*2) = 1*(1*2) (1*3) = (1*3) 2 = 2  ab, b = c (1*2)(1*2) = 1*(2*2) 3*3 = 1*2 3 = 3  ab, bc (1*2)(1*3) = 1*(2*3) 3*2 = 1*1 1 = 1 Note: the order of b and c in the above example is interchangeable.