Theorem 1   Let $l$ be a a non-vertical line in the Cartesian plane. Then for any two pairs of distinct points $P_1,P_2$ and $P_1',P_2'$ in $l$ we have $$m(P_1,P_2) = m(P_1',P_2')\,.$$

Proof. Let $P_1 = (x_1,y_1)$ , $P_2 = (x_2,y_2)$ and $P_1' = (x_1',y_1')$ , $P_2' = (x_2',y_2')$ be two arbitrary pairs of points in $l$ . Draw a vertical line $v$ (respectively $v'$ ) from $P_2$ (resp. $P_2'$ ) and a horizontal line $h$ (resp. $h'$ ) from $P_1$ (resp. $P_1'$ ) and let $Q$ (resp. $Q'$ ) denote the point where these two lines meet. If we temporally assume that $l$ is also not horizontal we have two nondegenerate right triangles $QP_1P_2$ and $Q'P_1'P_2'$ (see Figure 1).

These two triangles are similar and therefore we have that $$\frac{QP_2}{QP_1} = \frac{Q'P_2'}{Q'P_1'} $$ or equivalently since $Q$ (resp. $Q'$ ) has coordinates $(x_2,y_1)$ (resp. $(x_2',y_1')$ ) $$ \frac{\vert y_2-y_1 \vert }{\vert x_2-x_1 \vert} = \frac{\vert y_2'-y_1'\vert }{\vert x_2'-x_1'\vert}\,.$$ To conclude that \begin{equation} \label{eq:slopewelldfnd} \frac{y_2-y_1}{x_2-x_1} = \frac{y_2'-y_1'}{x_2'-x_1'} \end{equation}we argue as follows. First notice that $m(P_1,P_2)$ is symmetric with respect to $P_1$ and $P_2$ and therefore without loss of generality (exchanging the roles of $P_1$ and $P_2$ if necessary) we may assume that both denominators in () are positive, that is $Q$ (resp. $Q'$ ) is to the right of $P_1$ (resp. $P_1'$ ). We will show that then $P_2$ is above $Q$ exactly when $P_2'$ is above $Q'$ . To see this notice that by the fifth postulate of Euclid $P_2$ is above $Q$ exactly when the angle $QP_1P_2$ is acute, but this happens exactly when $Q'P_1'P_2'$ is acute, that is, exactly when $P_2'$ is above $Q'$ . Thus the numerators of the two fractions in () have the same sign. Thus Equation () holds.

We derived Equation () by assuming that the line $l$ is not horizontal (and therefore $P_1$ and $Q$ are distinct). However when $l$ is horizontal then the numerators of both fractions in Equation () are $0$ and the equation still holds. Thus the theorem is true for any non-vertical line.