Special relativity: the transpose inverse of Lorentz matrix

The exercise wants me to prove the relation:
$$Lambda_{mu}^{;;nu}Lambda{^mu}_{;alpha}=delta^{nu}_{alpha}$$
And then conclude that $(Lambda_{mu}^{;;nu})$ is the transpose inverse of $(Lambda^{mu}_{;;nu})$. The first part I did well and can conclude that
$$Lambda_{mu}^{;;nu}=(Lambda^{-1}){^mu}_{;nu}tag{1}$$
and, by definition
$$(Lambda^T)_{mu}^{;;nu}=Lambda{^nu}_{;mu}tag{2}$$
I’m trying to, somehow, put $(2)$ in $(1)$ to get the answer, but I’m not sure if it’s the right way or if assumed something wrong. Any help with it would be great.