Two runners start a race at the same time and finish in a tie. Prove t

Step 1
${\displaystyle f\left(t\right)=g\left(t\right)-h\left(t\right)}$ - the difference positions of two runners.
${\displaystyle g\left(t\right)}$, ${\displaystyle h\left(t\right)}$ are continuons and differentiable. So, ${\displaystyle f\left(t\right)}$ is too continuons and differentiable.
We are attempting to prove that at some point during the race the difference in velocities between the runners was 0.
Find the first derivative of the equation ${\displaystyle f\left(t\right)=g\left(t\right)-h\left(t\right)}$.
${\displaystyle f^{\prime }\left(t\right)=g^{\prime }\left(t\right)-h^{\prime }\left(t\right)}$
Consider the intreval ${\displaystyle (t_0,t_1)}$, where is the start time ${\displaystyle t_0}$ and the finish time ${\displaystyle t_1}$.
Is given that runners start a race at same time and finis in a tie.
So, ${\displaystyle g\left(t_0\right)=h\left(t_0\right)}$ and ${\displaystyle g\left(t_1\right)=h\left(t_1\right)}$.
We have: ${\displaystyle f\left(t_0\right)=0}$ and ${\displaystyle f\left(t_1\right)=0}$
We can apply the Rolle's theorem, because we have: ${\displaystyle f\left(t_0\right)=f\left(t_1\right)}$ and ${\displaystyle f\left(t\right)}$ is continuons and differentiable.
So, there exists some cc that such ${\displaystyle f^{\prime }\left(c\right)=0}$
${\displaystyle f^{\prime }\left(c\right)=g^{\prime }\left(c\right)-h^{\prime }\left(c\right)=0}$
${\displaystyle \Rightarrow g^{\prime }\left(c\right)=h^{\prime }\left(c\right)}$
For ${\displaystyle c=t}$, ${\displaystyle t\in (t_0,t_1)}$
${\displaystyle \Rightarrow g^{\prime }\left(t\right)=h^{\prime }\left(t\right)}$
Answer
Apply the Rolle's theorem.