Uniqueness of $V\rightarrow \frac{DV}{dt}$

Question

As picture below, although I have (1), I can't get the unique. Because
$$ \frac{DV_1}{dt}=\frac{dv_1^j}{dt} X_j +\frac{dx_i}{dt}v_1^j\Gamma_{ij}^k X_k =(\frac{dv_1^k}{dt} +\frac{dx_i}{dt}v_1^j\Gamma_{ij}^k )X_k $$ if $\frac{DV_1}{dt}=\frac{DV_2}{dt}$ , I have $$ \frac{d}{dt}(v_1^k-v_2^k)+\frac{dx_i}{dt}\Gamma_{ij}^k(v_1^j-v_2^j)=0 $$ But this don't mean $V_1=V_2$. Where is my mistake ?