 
The Brezzi theoremWe recall the Brezzi theorem, e.g. from [3].
THEOREM 1
Suppose a mixed variational problem in abstract setting
Proof. See, e.g., [4].
Find such that Assume that the following conditions are satisfied (A1) , , (A2) and with (A3) the LBB (LadyzenskayaBabuškaBrezzi) condition, i.e. such that
(A4) the  ellipticity, i.e. with
hold. We apply Theorem 1 to the variational formulation (VF1).
THEOREM 2
Suppose that
,
and that
is a piecewise constant function fulfilling
Proof.
We have to verify the assumptions of Theorem 1.
hold. (A1) It is the assumption, and . (A2) Applying Lemma 3 we get i.e., we can set , and i.e. . (A3) The LBB condition has been proved for the Stokes problem, cf. e.g. [6, Theorem 3.7]. There exists a constant with
(A4) We can prove ellipticity for the whole space , i.e. and . We remark that it may be useful to choose depending on the actual coefficients , e.g., .  
