Let:
[tex]\begin{gathered} P1=(x1,y1) \\ P2=(x2,y2) \end{gathered}[/tex]If the line is horizontal, then:
[tex]y1=y2[/tex]The slope is given by:
[tex]\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ since\colon \\ y1=y2 \\ m=\frac{y1-y1}{x2-x1}=\frac{0}{x2-x1} \end{gathered}[/tex]Zero over any quantity is always zero, so:
[tex]m=\frac{0}{x2-x1}=0[/tex]