to prove

a x b = b x c not equal to 0

post multiply by a

ax (a x b ) = a x (b x c)

a(a.b) - b (a.a) = b (a.c) - c (a.b)......(1)

now post multiply with c

c x(a x b ) = c x (b x c)

a(b.c) - b (a.c ) = b (c.c) - c (c.b) .....(2)

add (1) and (2)

(a+c) (a.b +b.c) = b ( a.a+a.c+c.c+a.c)

so a+ c = mb

where m = ( a.a+a.c+c.c+a.c) / (a.b +b.c)

hence proved