$RTP$: Show through the use of field axioms that $1 \in \Bbb{R}$ is unique.
Proof : The number 1 is unique in $\Bbb{R}$ if the following two properties are satisfied, i ) $a \cdot 1 = a$, and ii ) $a \cdot a^{-1} = 1$. Let $u=1$. Assume there exists a real number $v \neq u$ such that $$a \cdot v = a \cdot u$$ By the existence of a reciprocal and the associate law of a product $$(a^{-1}\cdot a)\cdot v=(a^{-1}\cdot a)\cdot u$$
Thus we have two results $$uv=uu$$ or $$vv=uv$$ Both lead to $u=v$, which, through contradiction, must mean that 1 is unique in $\Bbb {R}$. Is there a flaw in this proof?