I have some questions about this proof of uniqueness.

First we get that k1=j1.

Then repeating the same steps, we can show that k2=j2.

But the problem is how can we

**end** this? j_r and k_l are not necessarily equal (r and l are not necessarily equal).

To prove uniqueness by contradiction, we assumed at the beginning that

To be completely general,

**r and l are not necessarily equal**, so when we repeat your procedure of dividing by the leading term, at the end, something is going to be left over on one side of the equation but not the other. Say I assume r<l, then using your trick we can show that k1=j1, k2=j2, k_r=j_r, but we cannot say anything about j_(r+1), j_(r+2),...,j_l.