Proving Standard Inner Function Property 2

Consider $\overrightarrow{u},\overrightarrow{v_1},\overrightarrow{v_2}\in {\mathbb{R}}^n$ $\overrightarrow{u}=(u_1,\dots ,u_n)$ $\overrightarrow{v}=()$ $⟨u_1,v_1+v_2⟩$ $⟨(u_1,\dots ,u_n),(v_{11}+v_{21},\dots ,v_{1n}+v_{2n})$ $={\sum }_{k=1}^n(u_i{v}_{i2}+u_i{v}_{2i})$ $={\sum }_{i=1}^n{u}_i{v}_{1i}+{\sum }_{k=1}^n{u}_i{v}_{2i}$ $=⟨(u_1,\dots ,u_n,(v_{11},\dots ,v_{1n})⟩$ … didnt get the rest