Question:
prove that (a*b)*c = a*(b*c) if (c*a)*b = 0
Answer:
Given (a * b) * c = a * (b * c)
=> -c * (a * b) = a * (b * c) {since (a * b) * c = -c * (a * b)}
=> -{(c . b)a - (c . a)b} = (a . c)b - (a . b)c
=> -(c . b)a + (c . a)b = (a . c)b - (a . b)c
=> -(c . b)a = -(a . b)c
=> -(b . c)a = -(b . a)c
=> (b . a)c - (b . c)a = 0
=> b * (c * a) = 0
=> -{(b * c) * a} = 0
=> (b * c) * a = 0
=> (c * a) * b = 0 {since (b * c) * a = (c * a) * b}
So, (a * b) * c = a * (b * c) iff (c * a) * b = 0
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