 So, the last time we considered a real manifold and considered a projection and we add one variable and we defined the GFD category of enhanced in the shears. So, that is the shears, that GFD category of in the shears on this product space and we neglect the in the shears that comes from the base space. So, we construct all the object and then it becomes a monoidal category by the tensor product, a kind of tensor product or convolution and it has in the home and for morphism or the real spaces, we can define the direct image function and the inverse image function. And it behaves just as a usual theory and so, we can work in this category. Okay. So, now I want to introduce the notion of constructible shears, all constructible shears or in the shears. Okay. So, M via sub-analytic space. So, it means more or less locally M is isomorphic to the closed sub-analytic set subset of our M. So, in this case, we say that is a sub-analytic space. So, you in general, you want something which is an atlas when transition maps are locally sub-analytic. Yeah, yeah, yeah. So, or you can consider that is a ringed space and whose ring is a sub-analytic and continuous function on this sub-analytic set. So, that is a ringed space, R ringed space, which is locally isomorphic to the ringed space associated to a closed sub-analytic subset. Okay. So, then you can define a sub-analytic subset of M and so on. So, because that sub-analytic set is more or less the smallest sub-category that is invariant by taking the complements, finite union, finite and any image of a proper real analytic map is sub-analytic. And also local nature. So, you want something that is just locally sub-analytic. Yeah, that is a corollary. I think that is a corollary of those properties. I think so. Yes, I think so. So, that is a small subset, small subset, small family, smallest family of subsets, of subsets, of real, say, real analytics. Mindful. Mindful, say, M. And that is stable by taking R. You set finite. Ah, okay. It must be locally finite union. Otherwise, you can do something more, more complicated than you go to infinity. Yes, locally finite. Yeah. Then, I think it's okay. Okay. Anyway, so, we consider a category of R-constructible indices of M tends to R infinity. So, it means our object is R-constructible. It means there is a local finite covering, open covering, such that every UI is sub-analytic in the closure of R, with R by, say, T1R, and such that any homology group on each UI is locally constant of finite. So, now, definition. So, now you consider K, say, so, that is a enhanced indices. So, that is an element of, say, it is IKM, is R-constructible. If for any locally closed open subset U, and past sub-analytic open subset, there is, there is F in, I think it doesn't matter so much, but constructible shields, such that KU tensor, or pi inverse KU tensor, K is isomorphic to KM, KME tensor plus F. So, in the definition, you hold for everybody, but you may not exist. There exists, yes. Oh, no. Local, yes, over the right. Over the right. Oh, there you go. From the top line. Oh, yes, yes. And the locally finite is it? The locally finite is, okay, the locally M is at R, yeah. Yeah, yeah. I'm sorry. Yeah, yeah. Yeah, I thought, otherwise it does not make sense. Can't constructible shields, not in shields. Now, here on the left, you define, yes? To the left, there are shields. Yeah, that is a shift. Not in the shift. Not in the shift. Not in the shift. I hear, oh, no, I'm sorry, so that's a, yeah, yeah, I'm sorry. That is a shift, and so that is constructed from the shift. That part is important, otherwise, yeah. So that is a definition. And why it is useful? Yeah, it's shift, and it's a shift from M, not on M prime, seven, seven, eight, seven, eight, seven, eight, seven, M cross. Yes, yes. I'm sorry. That is your M cross, I'm sorry. So I have several proportions. So that is, so let M, the category of, it is a category, that's full subcategory of are constructed in the shields, enhancing in the shields. Then that is triangulated. So if you have a distinct triangle angles, and if two terms are constructed, then the other, the last term is also are constructed. And locality, I think I'm not right, explicit, but locally, if enhanced in the shift is are constructed locally, then that is are constructed globally. So why that is important? There are many things, but one is the following. So assuming that F, you have, so that is one of the reasons. Yeah, here it is not necessarily are constructively, but so, and support, say F prime and you send it. So that is compact. Then you can calculate home, KM, E, E plus F. So those are this term, F prime. So that is, that is E, KM. So that is just F, K prime and limit, you take an inductive limit, and that is the, it doesn't matter so much, KM. So considering the support, they are closed in M tensors? Oh, you can take that, yeah. So that closure is compact. The closure of the intersection? No, no, no, no, no, no, you can take an intersection, that is okay. That part is important. Anyway, so usually it is not easy to calculate home between the arbitrary industries, but in this case, so do other four shifts so you can calculate. Is the limit stabilized for some part? I don't think so. No, no, no, no, let me see. Because we consider compact, I, yeah, perhaps, yes, but I don't know. I think because we assume the constructive and that is compact, I think it should be, yeah, yeah, okay. So there are other things for constructive shifts, good things for constructive shifts, that in fact we use it later. So it is that behaves like the usual constructive shifts. And we need the following fact. It is stable by the duality function. So we define the duality function, okay, it is, oh, that's pi inverse m pi m. So that is a dualizing shift, topological dualizing shift, and you consider that. So theorem or proposition is, oh, yeah, that's it. I'll give many properties for our constructiveness. So if k1 and k2 are constructible, then there tends to be a product convolution and e1, so they belong again constructively. And so that is stable by the kinds of product or the convolution, and that is stable by the duality. So if k is are constructible, then its dual is are constructible. And the dual over dual, there's a canonical map, morphism to the pi dual, and that is a natural. In fact, that comes from the following fact, d e m k m e plus f, I think that is d that are constructible, sheaves, no, no, no, not in the sheaves, but construct one. Then that is tensor plus. So here, this one is a topological dual. So d f is the usual one, f plus r. So that is a very dear dual in the usual sense. And a is the antipodal map, x t goes to x minus t. So we know that the duality factor transforms the sense that are constructively sheaves, are constructively sheaves, so it must be are constructible. And by duality, it also follows from the fact that are construct sheaves are stable as a pi dual over the are constructively sheaves are constructively sheaves. What did you write below d m k m z over r? A sheaves. Are you mean this one? Antipodal map x t goes to x minus t. Antipodal? x t to x minus t. So you change the sign. Yeah, because plus is given by contravariant, that is home, so there is a sign change. Okay, so as a corollary, for example, in a usual shift case, I think that is a very, very no formula. So you consider the home function, the dual form function is the d i x t k 2 or so that is almost for k 1, k 2. So that is a usual way. And so they are stable, the property of are constructively sheaves are stable by the direct image, as direct image if for say, say f is proper and the inverse image. Okay, so those are the properties. So they are stable by two inverse images and proper direct images. Yeah, or a direct image by the proper map. Okay, I think that is the way. So now we want to consider the Riemann-Schilder problem for the regular case. And the idea is we work always in this category. And then more or less, that all the things are more or less similar way as the regular case. Okay, so first we discuss about the most typical irregular autonomic modules. So that is an exponential module. So we work on a complex manifold and Merlmark function with a pole and y. That is a closed hypersurface of x. You take that. And then we consider the following module, phi. So that is d over p such that p is 0, say outside y. So you can calculate it algebraically, so there is no problem. And you divide it. So that is a coherent left ideal of d. So we shall denote it by d. And we denote x phi. So that is d e phi star y. So that is by definition of x star y. So that is a shift of Merlmark functions whose poles are contained in y. Okay, so they are the autonomic. And that is as o x module, as almost generated by one element. If I correspond to the generator of this. Okay, so, you know, perhaps, let me see here, depends, right? Yeah, perhaps it's not a good but, yeah. So the 0 m is, so now you consider the drum in x, y. So that is dx where u is a complement of y. And so that is a in the shift for x in u, such that real part of phi x is smaller than a. So that is an open subset. So you consider local, the constant shift on it and you extend it to the outside. And because the art is increasing, so you, so that gives an induction system of sheaves and you take the ring. And dx is dimension x. So that is, so I think I'll not give the proof but I'll give some remark. Recall that drum x t y u phi x phi. So that is omega x t tensor dx. So d phi star y. And we can easily see that that is again equal to star y tensor dx d e star y. And then it is rather easy to calculate. So this part is t is e home or r e, yeah r e home c u. And this one goes through. So that is why here is c u. So it means that if you know this one outside y, then you can recover this one. So for melomorphic one, the drum complex is rather easy to calculate. So that is and yeah perhaps I think I'll not give the proof about that. Excuse me, do you need to solve the singularity proposed to resolve the singularity of phi to compute by push forward? We don't need it in fact. Yeah, well let me see. So the proof is we give to the one-dimensional case more or less. And in the one-dimensional case so one-dimensional case that is exponential x plus the solution to this one. So d over dx minus 1 and minus i 0. And for this part, so that is exponential x and exponential i y. And for this part, that is tempered. So you can transform to the usual one by multiplying exponential i y. So that is not important. Then for this one, so that is a well-known result of ordinary differential equations. I did it in my question. Ah, you did it. Okay. But finally, one needs the one act or one need or not? Perhaps not, but I don't have a question here because the we use the fact that all x or y, 10 part one is stable. So for example, we use the fact that if there's a map from x to y, this one is say f exponential y. So first we use such kind of things. I'm quite sure because that is based on the theory of temperature distribution or temperature hormone function. So I didn't check that if it doesn't. So in some sense it's a very long step. Okay. So that can be calculated in this way. Now, so now we work in an enhanced category. So we would define for it all in the enhanced category. So that is, I'm going to explain, enhanced tempered hormone function and so on. So first let us take a real analytic manifold and that is mp. Say p is p1r, a compactification of r and of course that is contained mp, p is so say j. So we first define the tempered distribution on the broader space m times r as an inverse image of next, we consider following shifts. So that is now and so that is d, t here. So that is a left dp module and that is a corresponding right module. So it means explicitly by the differential equation for dt that is the mp of infinity to d over dt minus 1 dv mp of infinity t and here d degree minus 1, here d degree is 0. So it corresponds to that complex. It turns out that the cohomology group, so that is a in-shift m times r infinity. So the cohomology is concentrated minus 1 so it means this morphism is subjective epimorphism and so that is equal to the kernel of this one shifted by 1. Okay. So that is a dp, dv and then we can prove the following see. So this one plus t is, so that is more or less the corresponding in-shift is stable. So we denote m, so that is corresponding element in ICM enhanced in-shift or m corresponding to the corresponding, other corresponding to dv, dv, t. So that is a, it corresponds to the distribution, the temporary distribution. That is an enhanced version of the temporary distribution. And so now you consider x, a complex manifold and then we define o xe. So that is a corresponding double complex. So that is on an xe 1030 xe tensor distribution e x. x you consider as a real analytic manifold and you can define this enhanced in-shifts and so that is a xc is a complex conjugate. So it is nothing but the usual double complex. No shift? I don't think so. No shift, double complex. So that is a double complex. So the, yes, so that corresponds to the, to the shift of homework functions or temporary homework functions. So for example, if x is point, then o x or point e, so that is a dv point e and that is a point e. So that is a limit. So in this case that is a constant shift, the corresponding constant. Okay, so let me see. So the idea is we replace all the same to the enhanced version. For example, we define enhanced drum for d module. So that is on an xe xm or solution is xm o xe where when xe is on an xe, so you just tensor invertible c for the highest degree differential, differential forms. So that is a left, yes. So there is a left action of d and, you know, there is a right action of d. So you can take the home and that is it. And solution also. Of course that is isomorphic to ram x. Let me see. There is some shift, up to shift. So there is a drum and solution function in the enhanced category. So that is what we use. So we replace all the operations to this category. and m to this that means R infinity. But no no, omega x extended its object on m tensor. Yes, so I think, I don't know, first, yes, that is, first, better to write pi inverse, or it depends, yeah, omega x, where pi is a x tensor, R, better to write it. So, the four-ingutal almost trivial, m x, m is stable, because stable means there are many characterizations, but c x, e, tensor, m x, e, m is m x, m, or e, m. And that comes from the fact that, I didn't, let me see, I'm confused, oh yeah, yeah, yeah. So, from this, it implies that dve, m is stable. So, then, o xe is stable again, and dve is stable again. So, they are stable object. Okay, so, so, now, we want to study the drama of the phonomic system or, or those kind of things. And the strategy is very similar to the regular case. I think in the lecture of Shapiro, you learn that the, we reduced the, those properties to the simple case with normal crossing singularity case. And in the usual, in the irregular case, that is similar. So, in the regular case, so, any module is more or less even by the d over, you can reduce those kind of things. And in the irregular case, it's a little bit more complicated. I think that I explained in one dimensional case. And in a stable dimensional case, that we have to use the result of Mochizuki and Keduraya that I'm going to explain now. So, so, in regular case is a little bit complicated. So, I'll explain. So, recall that, I think that is already done by Shapiro. So, you take a complex manifold, so, normal form of irregular phonomic modules. So, you take a complex manifold and d normal crossing hyper surface. Then, you can define a real blow up around the fiber positive. I think that is explained by Shapiro. And it is contained in the ambient space. And local picture is x is minus r. So, that is a local picture. And d is still inside, normal crossing hyper surface. And x is and c tilde. And for example, that tz such that is 1, c. And there is a map to c, c, z, z. And c, tz, and so on. So, that is a local, I think that is already explained by Shapiro. And so, that is isomorphic. The complement of d is isomorphic to this one. And x tilde is a closure of x tilde positive. And x tilde total is a germ of the of a real analytic manifold contained x tilde. And so, say, j. And we call that a is a homomorphic function defined on x tilde or x the complement of d such that f is temporary. I think that is already explained. So, I will not prove. And so, it contains pi inverse of x. So, that is algebra. It contains pi inverse of x. And you define dx tilde a. So, you just tx. So, that is a shape of differential operator with coefficient a. And m a is 8 times say pi inverse of x. m 4m is dx. So, that is a corresponding dx a. So, now, I want to give the corresponding irregular case of what is the corresponding one, the canonical one. So, that is called normal form. So, this is an autonomic dx model has a normal form along d. So, we assume that we have a compressible for the x and the normal crossing divisor d. If I do leave, it satisfies certain condition. First condition is m is monomorphic. And m is, I don't know how to say, m is a monomorphic connection. So, there are many ways to say it, but it is a coherent or x-module. So, it must be a locally free or x-module of finite rank. And the differential gives a normal connection with positive d. So, that is one condition. And another condition is a lot of complicated. So, more or less locally on x tilde, that is a orthogonal to the exponential modules. So, that is a condition. Not precisely that is for anything. So, you take, for any point in pi inverse d, you can take neighborhood of pi over p in x and neighborhood tilde pi i. So, that is a monomorphic function with pole on d, pi. So, that is a finite number of finite family of monomorphic functions, such that m a, m a is this one, you pull back to x tilde and it tends to a. So, that is dm-module, m-module. And you consider locally on x tilde. So, you district it to a neighborhood of p. Then that is, so that's a finite sum. So, it means more or less if you go up to x tilde and you consider in the category of a-modules, you permit a, then that is equal to the exponential modules. So, that is a normal function. So, we replace the, that bring that one to the, this one to the, this one. Okay. So, in general the normal function is okay. So, in terms of, so that could lie, the normal function, can you state it? Can you state the normal, the, you said there is a theorem of Shizuki. Could you read? Yeah. So, I think I saw it in some talks. Yeah. Can you say exactly what the theorem is? In other words, I will tell you. Yeah, yeah. Yes, yes. So, that part is important. Yeah. And in fact, that is a little bit complicated. So, yeah. So, what happened is, so first Mochizuki proved in an algebraic case in a formal setting and then Kedura proved similar normalization program for, in a analytic case and then Mochizuki proved another one. So, so that is A. A means more or less we consider asymptotic expansions or something like that. So, you remember that. So, in a one-dimensional case, there are two statements. One is every, one has a formal solution of some form. And the second one is in fact, there is a solution. But that is asymptotically equal to it. And you employ A, it means more or less asymptotically equivalent. And formal and asymptotically is a little bit different. And there is a classical technique to go from formal to... Okay, but is there a, the formal theorem is that if you have... A form? I think I'll explain a little bit later. Yeah, we have 15 minutes, please. Wait a minute. We shall start at 345. Okay. So, here we define the normal form along a normal crossing divisor. So, it is a normal connection outside D. That is asymptotically more or less asymptotically equal to the exponential modules upstairs, I mean, in the real blow up. But it is not enough, as I think it occurred in the question of rubber. So, we define another notion. M has a caseinormal form, D. So, it means, so locally take a coordinate, such that D is given by 0. And you take a covering, say, you take MK, and you take the covering, say, to, so the cover, the usual covering. So, you consider ramification along the divisor, such that star M. So, you pull back. M has a normal form along the inverse. So, that is a covering. Okay. So, then we can state the result of Mochizuki and Kedryer in a asymptotically form. Let me see. I think about this. So, you consider X a complex manifold, and M is a holonomic TX model. And you take a point. Then, after the blow up, so, then there is an open neighborhood over X and X map, projective map, and Y calls the subspace, Y calls the subspace. It doesn't matter so much. And X prime and say F, F inverse Y is normal crossing, hypersurface. And you pull back M, and you look as D. So, it has a normal form. I don't know. I'm sorry. F is one-to-one outside of Y. F is identity outside of Y, the snap F. Now, X prime has the same dimension. D is a, ah, I'm sorry. I'm sorry. Yes. So, X prime over D is Y. I'm sorry. I forgot condition. So, that's a blow up. So, here, yes, the X prime is a, ah, okay, you like it. So, that is locally on X. I think, so, I'll comment something. So, first, so, it's Mochizuki proved in algebra case, and you can prove it in a light case, but not in that form. But in the following form, you have X, and you consider the compression of X tilde. So, that is a compression, more or less a compression of X along D. And you can consider DX. So, that is OX, DX. Then, I think that is DXM. So, so, we change this condition with this condition, DX. So, there's a FI and so on. So, ah, yes. So, that's more, ah. No, no, no, I'm sorry. We need something else. So, it's a little bit, so, plus some log term and, and that I learned at the time. So, OR plus L, right, L, I. So, that is regular. Say, regular local system, regular holonomic. Yeah, ah, yeah, yeah, yeah, yeah. Yes, I'm sorry. So, so, I mean, we replace the condition of normal by this condition, and then what K-Tri and Mochizuki first proved that not formal one, but in a, ah, not in a simple sense, but in a formal sense and has a case I know. I mean, after. Do you know the whole device or there are several completion. You can either complete each point or for a spine open. Ah, so, that is local. That is a local one. Yeah, yeah, yeah. What on T or? That is a local condition. So, you take it as a spine open and you complete it on T or you get the stratified completed normal. No, no, no. Because he says, he says that it's a completion along stratile. I'm not sure exactly what is this. No, that is okay, but the condition, yeah, it's, no, that is a definition. That is a definition and you change normal, I mean, this condition with this condition, and you define case and normal in this way. Then what they first proved is in algebra case by Mochizuki and the, you know, not case by K-Tri, this statement. So, it's a local. The CI are defined where? Are they defined after the blow up or after? After the blow up, of course. Yes, yes. In other words, the idea is that. Yeah, CI is upstairs. CI is not malmorphic. Not, it is malmorphic, but it is, no. Absolutely defined locally upstairs. Yeah, yeah. I think we don't, I think there is, but it's a, yeah, upstairs or upstairs. So, this is just a definition, then you say there's a group, no. So CI is in, well, they are team angels, no, but the formal version. Yeah, yes. In a formal version, so for any point in the, you can find the neighborhood such as on that neighborhood, if you, after the tensoring, that is as one of the two. That, for some of the, yes, we needed it. And the case of normal form is after the ramification, it becomes normal form. But X hat is not the completion of X along D. It's a little bit complicated. It's along the strata. I don't think so. No, I think it's okay. I think so. I think by that it's okay. Okay, so you, and this is, let us say, so this means that it's the ring space obtained by the annual stream rate of infinitesimal neighborhoods of D considering, let us say, the analytic case, the user topology goal in the algebraic, you do it in which analytic case or in the Are you making this one? You have all the, the complex one, so it's the complex topology. Yeah, so, so that is a, a sympathetic version of this one. And it will most likely have proved that the formal statement implies the asymptotic statements. So then a sympathetic statement, so that is given in terms of asymptotic expansions. Okay, so then what happens? Then we have, yeah, then I think that is the repetition, right? So many statements for irregular D modules can be reduced to the normal form case. So that is a similar to the irregular case. So we consider a statement for irregular, for for an economic DX module and, and we assume the conditions F, then it's true for any. So that is a similar to the, so once A is low quality, I will not repeat it. So if the statement is true locally, then it is true for M. And, and not for economic DX, for M, sorry, so MN and stable by this thing is two ions, so this thing is two ions and PXM, PXM double prime implies PXM. And stable by direct summon implies PXM and F is projective and M is X and good module. So more or less that is has a global defined filtration. Then PXM implies direct images and the last condition is M has a normal form, then PXM holds. So usually those proportions are, we can prove some more. And so more or less this lemma says by some procedure we can reduce to a normal form case. So that is enough, this one. That's a quasi normal just to compare the skin line. Ah, yeah, yeah, so we, so of course actually it's a quasi normal case, but TP star M is normal. And then P star M is true and M is a direct summon, so by using those ones. So quasi normal case is also true. Then by multi-state layer, you'll find the probe, then the F star M, so that has a case of normal form, so the statement P is true for this one. Then you go to the direct image, then M is again. Because you need some induction of the support, because if it is not supported you have to resolve the signal. Yeah, that's right, yeah, yeah, yes, and because of this one you can add some more, yeah. So that is the way. So the strategy is by using this lemma or by using the KDL multi-state theorem, we deduce the statement to a normal form case. And we have to do it. So that is the strategy. Okay, yeah, so in order to do that, at least we have to know the following one. So assume that phi is a memory function whose pole is contained in a normal coaxing device at D. Then the term enhanced term of the exponential model. So what is this one? So that is, I think I erased already perhaps. So that can be calculated easily and that is given by dx. dx is a dimension and u is a x minus k, the complement. Or there are many ways to write it. So for example, cx e plus e home minus u is e to the power of c dx, or you can put it inside. There are many ways. So I think it will have t equal to 1. So that is a typical case, so the term. Okay, so what to start? Yeah, for example, so that is a theorem. We put those kind of theorem, first theorem. Assume that m is a polynomial, then you have the enhanced term. And that is, in fact, our construct. We call that, that is our constructor means locally that is of this form and f is our constructor in cx. So it comes from a shift. Yes, for example, in this case, it is a case that turns out. And this one is our constructor. Okay, so how to prove it? I don't know how to explain it yet. So it's dp of cx cross 7p, yes. Ah, yes, I'm sorry. Yes, so we use this lemma and we use it to the normal case. I think I will not state for those ones. And we consider the statement that pxm such that run x, m is our constructible. And we prove those properties. And the problem is, we have to prove, we use to the normal form case. And for those ones, I think we can see it rather easily. With the help of the theory of temperature revolutions and temperature homework functions. Okay, so how to prove that statement in the normal form case. Then in the normal form case, if you go to upstairs, x tilde, then m has a normal form. So we have to compare drum on the upstairs, x tilde, and on x. And that is done in the following way. So you have x tilde and x, and so that is y tilde pi. And that total space, that is a real analytic neighborhood of x tilde. And similarly, you have infinity, infinity, and so we defined drum, enhanced drum on x. And so now I'm going to define enhanced drum on x tilde. And we compare drum upstairs on x tilde and drum downstairs on x. So that is a strategy. And so we do the similar thing to define drum, usually enhanced drum. So it means, you remember that we first key. So we define plus e home, so that is e i. So that is a template distribution defined on this space. But of course, that is the dx A is a differential operator on x with a coefficient in A. And A acts, of course, this template distribution because A is the chief of homework function tempered along the boundary. So that is a double action, dh power connection. So that is FD source. And so you define x tilde e. So the address in a way, the corresponding, I did the drum. Yeah, that's it. I was wrong. I think the answer is correct. Yeah, that is your question. I'm sorry. So that is dx tilde. Oh, no, no, no. Let me see. And so that is our t and dbex tilde say. So that is d e t tensor p t and o x tilde e. I think, so it must be this one at home. I think I did something wrong. So that is e with a coefficient in dx tilde a. So dx tilde a acts on o x tilde a. I think I think I was wrong with the definition of I think all are erased. I think we define what I defined in this way. So that is false. It is, I think if we say that, we need the logic dx minus dx. Minus, I'm sorry. So it all homo is correct. So the correct answer is home. So that is correct. Okay. Anyway, so we define in this way. So that is a counterpart of o x e in the upstairs. And so that has a dx module structure. And you go up by pi upper shurik. Then it is isomorphic to o x tilde t tilde e. O x tilde e is a complex of dx tilde a module. And so, so the advantage of this one is this one, apparently this one does not have a dx a action. But since they are isomorphic, this has a dx a action. So you can go to the dx a. So that is an idea. And so, so that is a consequence. If you go downstairs, that is o x. So those are two relations. So, so this proportion says, I think that part is important. So I'll repeat it. So this one has originally only a dx action. But if you go up onto x tilde, then it admits a action, a symmetric action also. So that is an advantage. Okay. So now I'm going to prove this theorem. And by that deficit, I mean that you are using the last round. We can reduce to the all construct. Oh no, no, no, no, no, no, no, no, no, no. The normal form case, let me see. It's still 5 minutes. Yeah, not too short, but it's a little, yeah. So you define E, M. So that is, say, MA, so that is A tensor dx tilde E MA is drum EXM, because of this. So now the strategy is the following. And of course, the drum EXT tilde E MA is isomorphic to upstairs. And so you go up and down by using these two functions. It comes from those two isomorphisms. Okay, so the strategy is the following. So you have a normal form M, you go up, so that is isomorphic to MA. And MA is local isomorphic to the exponential module MA. And so drum EXT tilde E MA is local isomorphic to drum EXT tilde exponential of E A is isomorphic to E pi drum EXE exponential of E. And this one, we know it already, and that is R-constructible. And so it's inverse image, upper shrink, that is R-constructible. And so that is R-constructible. And so that is R-constructible, because R-constructivity is a local notion. So that is local isomorphism, but that is locally R-constructible, so globally R-constructible. So if you go down, so that is R-constructible.