 Then what you need to do is that you, you look at, you're going to take a limit on both sides and use the DV-curve lemma here. Okay, I'll just explain here. First of all, let's look at the right hand, I mean the left, the left hand side here. It's a product here. Both has weak limit. This has a limit actually by the assumption. The limit is, the weak limit is h. This one has the limit, weak limit, because of the, the first term I have here. All right, that's because once you take the gradient in, this linear function becomes a constant, constant of course is the periodic function, and then you take a gradient here, which will have an epsilon come out on the denominator, cancel this out. Then you have an xk star of x over epsilon L. xk star is a periodic, scaled, you have a limit by that theorem. So both has a limit, weak limit, and then you have to verify whether you have diverges at the curve here. So I guess here, because this is in a gradient form, the curve is 0, and here the diverges, converges, well, converges by the assumption of the theorem. So the left hand side converges to something. You can actually tell this is converged to h by definition here, and this one will converge to the average of the periodic function of, the average of this periodic function is 0, b is the corrector to define, the mean is 0, and this one when you take the gradient is a constant here, becomes just the gradient of xk, so that's, and then you multiply the test function phi here. That's half on the right hand side. Okay, on the left hand side, on the right hand side, you're going to do the same thing by kind of switch order here. Okay, again, both term has weak limit. This has a weak limit by the assumption. This has a weak limit because each of this is a periodic function. The product is a periodic function. Scaled. It has a limit, and then diverges of this gradient is 0, and no, sorry, the core of this gradient is 0, and the diverges of this term is 0 because this kai star is the solution of that equation for a star. Okay, so the right hand side also converge to, this one will converge to the gradient of u, and this will actually converge to something related to well, the limit here using, it converges to the limit of this one, which actually is a 0. You have a star there. Okay, so at the left hand side, the right hand side actually converges to a naught star and the gradient of xk and phi dx. Okay, so that's that's the idea. I'm running out of time. I probably spent just a little bit of time tomorrow to finish the proof, but that's all the main idea. It's already here. Okay, I'll stop here today. Thank you. Yeah, yeah, yeah, because it's a limit of a constant. It's a, yeah, so each AL hat is constant, the limit is still constant. Yeah. It's all, it's all, it's still elliptic. Satisfy the same lower bound. Thank you.