Valid reason to prove the dis-associativity of ($\mathbb R, -)$.

Question

Is this a valid proof?

Let $a,b,c$ $\in$ $(\mathbb R, -)$. We want to prove that $(\mathbb R, -)$ does not have an associative property. Assume that it is associative such that: $a-(b-c) = (a-b)-c$.

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$a-(b-c) = (a-b)-c$
$\Rightarrow$ $a+c=(a-b)+(b-c)$
$\Rightarrow$ $c+a=(a-b)+(b-c)$

So $c=a-b$ and $a=b-c.$
But $c=a-b$ is equal to $a=c+b$, which is a contradiction to the previous statement.

Hence $(\mathbb R, -)$ does not have an associative property.

Answer 1

$c+a=(a-b)+(b-c)$ so $c=a-b$ and $a=b-c.$

No, that is wrong. You know that $1+3 = 2+2$, right? But does it mean that $1=2$ and $3=2$?

The real solution is simple. Take $a=b=c=1$ and evaluate both sides.

Answer 2

We have $$\begin{align} (1-1)-1&=0-1 \\ &=-1 \\ &\neq 1 \\ &=1-0 \\ &=1-(1-1),\end{align}$$ so $(1-1)-1\neq 1-(1-1)$. Thus $(\Bbb R, -)$ is not associative.