Multiplication using Parabola

Theorem
Given the function $f(x) = x^{2}$ and two points $(A,B)\in \{(x,f(x)):x \in \mathbb{R}\}\wedge A_{x} \leq B_{x}\wedge (A_{x} \neq 0 \vee B_{x} \neq 0)$ Then the line segment $AB$ will have a y-intercept of $-(AB)$.

Proof
First it is clear that $f(A_{x}) = A_{x}^{2}$ Which defines the y coordinate of A, the same logic follows for B.

It follows that the slope of the line segment that joins $AB$ is defined:

$$ m = \frac{B_{x}^{2}-A_{x}^{2}}{B_{x}-A_{x}} $$

The numerator being a difference of squares it can be easily simplified to:

$$ m = B_{x} + A_{x} $$

Now by choosing a point A or B sub into the equation from the line $y=mx+b$ and solve for the y-intercept $b$.

$$A_{x}^{2} = (B_{x}+A_{x})A_{x} + b $$ $$-A_{x}b_{x} = b$$

Q.E.D