Prove: if $a\times b=a\times c$ then $a || b-c$ when $a\neq 0$
So we have $$a\times b=a\times c\iff a\times b-a\times c=0\iff a\times(b-c) =0$$
So $$0=a\times(b-c)=|a|*|b-c|\sin \theta $$ Therefore $$ \sin \theta\Rightarrow\theta=0,\pi$$
But what happen if $b=c$ can we still say that $a || b-c$?
That all depends to your definition of parallelity of the zero vector. A definition may say that the zero vector is not parallel to any vector since it actually is a point and the parallelity of a point and a line isn't well defined.
Another definition is as follows:
Two vectors are parallel only if they have the same direction, say, their angles are equal with any given line.
Based on this definition, regardless of the magnitude of the zero vector, since the angle of it can be any real number then is parallel to any other vector, yet this definition has a flaw. If two vectors are parallel pairwise with a third vector, then the two themselves are also parallel. The latter definition breaks this property. Anyway, you must clarify your definition of parallelity of the zero vector.