1.11 If $\forall a \in \mathbb{F}. \: \exists b, c \in \mathbb{F}. \quad a+b = a+c$, then $b=c$

Proof:
\begin{align*}
b &= b + 0 && \text{(by 1.3)}\\
&= b + (a + (-a)) && \text{(by 1.4)}\\
&= (b + a) + (-a) && \text{(by 1.2)}\\
&= (a+b)+(-a) && \text{(by 1.1)}\\
&=(a+c)+(-a) && \text{(by the assumption)}\\
&=(c+a)+(-a) && \text{(by 1.1)}\\
&=c+(a+(-a)) && \text{(by 1.2)}\\
&=c+0 && \text{(by 1.4)}\\
&=c && \text{(by 1.3)}\\
\end{align*}