Let us consider $\Phi(\mathbf{x},\mathbf{y})$ which is Lipchitz differentiable. I am wondering if the following expression is whether true or not:
$$\Phi(\mathbf{x}_2,\mathbf{y}_2)-\Phi(\mathbf{x}_2,\mathbf{y}_1)-\langle \partial_y \Phi(\mathbf{x}_2,\mathbf{y}_2), \mathbf{y}_2-\mathbf{y}_1 \rangle = L \lVert \mathbf{y}_2-\mathbf{y}_1 \rVert,$$
where $L$ is the Lipchitz constant. Please also let me know any helpful study material in this regard.