Originally Posted by

**kingwinner** $\displaystyle

2^{\jmath_1}+2^{\jmath_2}+\cdots +2^{\jmath_r}=2^{k_1}+2^{k_2}+\cdots+2^{k_\ell}

$

WLOG, I assume r<l, then we can show that j1=k1, j2=k2, j3=k3, ..., j_r=k_r, and we can subtract 2^j1 from both sides, then subtract 2^j2 from both sides, then subtract 2^j3 from both sides, ..., then subtract 2^(j_r)from both sides. I understand everything up to here. What's giving me trouble is the justification AFTER this.

Then we will have some terms left over on the right hand side, namely 2^[k_(r+1)], 2^[k_(r+2)],..., 2^(k_l), and we can't use the same arguments.