Knowing that lines $y=a_1 x+b_1 x^2$ and $y=a_2 x+b_2 x^2$ coincide in every point we can infer that $a_1=a_2$ and $b_1=b_2$. This inference is self-evident for me, but I don't know how to prove it formally.
How to prove this inference formally, using predicate logic? It would be nice to receive answer in terms of Fitch notation.
As far I understand $ \left\{ \begin{array}{c} y=a_1x+b_1 x^2 \\ y=a_2x+b_2 x^2 \end{array} \right. $ stands for crosspoint of two lines.
If the $y_1=a_1x+b_1x^2$ and $y_2=a_2x+b_2x^2$ coincide in every point, this means $y_1=y_2,\forall x\in\mathbb R$.
Let $x≠0$, then we have
$$\begin{align}&a_1x+b_1x^2=a_2x+b_2x^2 \\ \implies &x^2(b_2-b_1)+x(a_2-a_1)=0\\ \implies &x\left(x(b_2-b_1)+(a_2-a_1)\right)=0 \\ \implies &x(b_2-b_1)+(a_2-a_1)=0,\\ \implies &x=\frac{a_1-a_2}{b_2-b_1},~\text{if}~b_1≠b_2 \end{align}$$
But, that means if $x≠0$, then the coincide is at only one point, which gives a contradiction. Therefore, we deduce that
$$b_1=b_2$$
This means, $$a_1=a_2.$$