Let $E$ be an affine space attached to a $K$-vector space $T$, $a\in E$ and $u:E\rightarrow E$ an affine bijection. There exists a unique affine bijection $u_1:E\rightarrow E$ such that $u_1(a)=a$ and $t\in T$ such that $u=h_t\circ u_1$, where $h_t:x\mapsto t+x$.
The author suggests $t:=u(a)-a$ and $u_1:=u^{-1}h_t$. This implies that $u$ should be equal to $h_tu_1=h_tu^{-1}h_t$. But I cannot prove it. Let $v$ be the unique $K$-linear mapping such that $u(s+x)=v(s)+u(x)$ for all $x\in E$ and $s\in T$. Then for $x\in E$ we have $$(h_tu^{-1}h_t)(x)=h_t(u^{-1}((u(a)-a)+x)))=(u(a)-a)+u^{-1}((u(a)-a)+x);$$ the latter is equal to $(u(a)-a)+v^{-1}(u(a)-a)+u^{-1}(x)$. How can I get from this to $u(x)$?