Four-dimensional generalization of the equation $vec{D} = epsilon vec{E}$

In "The electrodynamics of moving dielectrics" (p. 262) Landau and Lifshitz have defined four-tensors and four-vector

$F^{mu nu} =
begin{pmatrix}
0&-E_x&-E_y&-E_z
E_x&0&-B_z&B_y
E_y&B_z&0&-B_x
E_z&-B_y&B_x&0
end{pmatrix}$

$H^{mu nu} =
begin{pmatrix}
0&-D_x&-D_y&-D_z
D_x&0&-H_z&H_y
D_y&H_z&0&-H_x
D_z&-H_y&H_x&0
end{pmatrix}$

$u^{mu} =
begin{pmatrix}
frac{1}{sqrt{1-v^2/c^2}},&frac{vec{v}}{csqrt{1-v^2/c^2}}
end{pmatrix}$

Then they write

From this four-vector and the four-tensors $F^{mu nu}$ and $H^{mu nu}$ we form combinations which become $vec{E}$ and $vec{D}$ in a medium at rest. These combinations are the four-vectors $F^{lambda mu}u_{mu}$ and $H^{lambda mu}u_{mu}$; for $vec{v} = 0$ their time components are zero and their space components are $vec{E}$ and $vec{D}$ respectively. The four-dimensional generalization of the equation $vec{D} = epsilon vec{E}$ is therefore evidently
$$ H^{lambda mu} u_{mu} = epsilon F^{lambda mu}u_{mu} $$

How is that evident?

I understand that with the assumption $vec{v} = 0$ that generalized equation gives us $vec{D} = epsilon vec{E}$. But couldn’t that just be a lucky coincidence?

Or is the reasoning something along these lines: "We found this equation, it gives good results, all the experiments work with that too, so it’s probably right." ?