165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
168 define |
168 define |
169 \[ |
169 \[ |
170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{k\delta_i}(|p|). |
170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{4^k\delta_i}(|p|). |
171 \] |
171 \] |
172 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
172 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
173 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
173 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
174 the size of the buffer around $|p|$. |
174 the size of the buffer around $|p|$. |
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175 (The $4^k$ comes from Lemma \ref{xxxx}.) |
175 |
176 |
176 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
177 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
177 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
178 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
178 = \deg(p) + \deg(b)$. |
179 = \deg(p) + \deg(b)$. |
179 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
180 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
190 |
191 |
191 As sketched above and explained in detail below, |
192 As sketched above and explained in detail below, |
192 $G_*^{i,m}$ is a subcomplex where it is easy to define |
193 $G_*^{i,m}$ is a subcomplex where it is easy to define |
193 the evaluation map. |
194 the evaluation map. |
194 The parameter $m$ controls the number of iterated homotopies we are able to construct |
195 The parameter $m$ controls the number of iterated homotopies we are able to construct |
195 (Lemma \ref{mhtyLemma}). |
196 (see Lemma \ref{mhtyLemma}). |
196 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
197 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
197 $CD_*(X)\ot \bc_*(X)$ (Lemma \ref{xxxlemma}). |
198 $CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{xxxlemma}). |
198 |
199 |
199 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
200 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
200 Let $p\ot b \in G_*^{i,m}$. |
201 Let $p\ot b \in G_*^{i,m}$. |
201 If $\deg(p) = 0$, define |
202 If $\deg(p) = 0$, define |
202 \[ |
203 \[ |
216 V^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
217 V^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
217 \] |
218 \] |
218 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
219 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
219 We therefore have splittings |
220 We therefore have splittings |
220 \[ |
221 \[ |
221 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
222 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
222 \] |
223 \] |
223 where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$, |
224 where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$, |
224 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
225 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
225 $e' \in \bc_*(p(V))$, and $e'' \in \bc_*(p(X\setmin V))$. |
226 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
226 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
227 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) |
227 near $\bd V)$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
228 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
228 unambiguous.) |
229 unambiguous.) |
229 We also have that $\deg(b'') = 0 = \deg(p'')$. |
230 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
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231 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
230 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
232 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
231 This is possible by \nn{...}. |
233 This is possible by \nn{...}. |
232 Finally, define |
234 Finally, define |
233 \[ |
235 \[ |
234 e(p\ot b) \deq x' \bullet p''(b'') . |
236 e(p\ot b) \deq x' \bullet p''(b'') . |
304 Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the |
306 Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the |
305 $G_*^{i,m}$. |
307 $G_*^{i,m}$. |
306 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
308 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
307 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
309 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
308 Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
310 Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
309 Recall that $h_j$ and also its homotopy back to the identity do not increase |
311 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
310 supports. |
312 supports. |
311 Define |
313 Define |
312 \[ |
314 \[ |
313 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
315 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
314 \] |
316 \] |
315 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
317 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
316 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
318 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
317 (depending on $b$, $n = \deg(p)$ and $m$). |
319 (depending on $b$, $n = \deg(p)$ and $m$). |
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320 \nn{not the same $n$ as the dimension of the manifolds; fix this} |
318 |
321 |
319 \begin{lemma} |
322 \begin{lemma} |
320 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
323 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
321 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
324 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
322 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
325 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
325 |
328 |
326 \begin{proof} |
329 \begin{proof} |
327 Let $c$ be a subset of the blobs of $b$. |
330 Let $c$ be a subset of the blobs of $b$. |
328 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
331 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
329 and all such $c$. |
332 and all such $c$. |
330 (Here we are using a piecewise smoothness assumption for $\bd c$). |
333 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
331 |
334 the fact that $\bd c$ is collared.) |
332 Let $r = \deg(b)$. |
335 |
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336 Let $r = \deg(b)$ and |
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337 \[ |
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338 t = r+n+m+1 . |
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339 \] |
333 |
340 |
334 Choose $k = k_{bmn}$ such that |
341 Choose $k = k_{bmn}$ such that |
335 \[ |
342 \[ |
336 (r+n+m+1)\ep_k < l |
343 t\ep_k < l |
337 \] |
344 \] |
338 and |
345 and |
339 \[ |
346 \[ |
340 n\cdot (3\delta_k\cdot(r+n+m+1)) < \ep_k . |
347 n\cdot ( 4^t \delta_i) < \ep_k/3 . |
341 \] |
348 \] |
342 Let $i \ge k_{bmn}$. |
349 Let $i \ge k_{bmn}$. |
343 Choose $j = j_i$ so that |
350 Choose $j = j_i$ so that |
344 \[ |
351 \[ |
345 3\cdot(r+n+m+1)\gamma_j < \ep_i |
352 t\gamma_j < \ep_i/3 |
346 \] |
353 \] |
347 and also so that for any subset $S\sub X$ of diameter less than or equal to |
354 and also so that $\gamma_j$ is less than the constant $\eta(X, m, k)$ of Lemma \ref{xxyy5}. |
348 $2n\gamma_j$ we have that $\Nbd_u(S)$ is |
355 |
349 \end{proof} |
356 \nn{...} |
350 |
357 |
351 |
358 \end{proof} |
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359 |
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360 In the next few lemmas we have made no effort to optimize the various bounds. |
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361 (The bounds are, however, optimal in the sense of minimizing the amount of work |
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362 we do. Equivalently, they are the first bounds we thought of.) |
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363 |
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364 We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
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365 some metric ball of radius $r$. |
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366 |
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367 \begin{lemma} |
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368 Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
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369 Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
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370 \end{lemma} |
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371 |
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372 \begin{proof} \label{xxyy2} |
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373 Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
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374 Note that $\Nbd_a(S) \sup B_r(y)$. |
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375 Simple applications of the triangle inequality show that $\Nbd_a(S)$ |
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376 is star-shaped with respect to $y$. |
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377 \end{proof} |
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378 |
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379 |
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380 \begin{lemma} \label{xxyy3} |
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381 Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) |
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382 of $k$ metric balls of radius $r$. |
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383 Then there exists a neighborhood $U$ of $S$ such that $U$ is homeomorphic to a disjoint union |
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384 of balls and |
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385 \[ |
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386 \Nbd_{2r}(S) \subeq U \subeq \Nbd_{4^k r}(S) . |
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387 \] |
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388 \end{lemma} |
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389 |
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390 \begin{proof} |
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391 Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
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392 By Lemma \ref{xxyy2}, each $\Nbd(S_i)$ is homeomorphic to a ball. |
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393 If these balls are disjoint (always the case if $k=1$) we are done. |
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394 If two (or more) of them intersect, then $S$ is contained in a union of $k-1$ metric |
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395 balls of radius $4r$. |
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396 By induction, there is a neighborhood $U$ of $S$ such that |
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397 \[ |
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398 U \subeq \Nbd_{4^{k-1}\cdot4r} . |
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399 \] |
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400 \end{proof} |
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401 |
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402 \begin{lemma} \label{xxyy4} |
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403 Let $S \sub \ebb^n$ be contained in a union (not necessarily disjoint) |
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404 of $k$ metric balls of radius $r$. |
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405 Then there exist neighborhoods $U_0, U_1, U_2, \ldots$ of $S$, |
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406 each homeomorphic to a disjoint union of balls, such that |
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407 \[ |
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408 \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) |
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409 \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) |
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410 \subeq U_2 \subeq \Nbd_{4^{3k} r}(S) \cdots |
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411 \] |
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412 \end{lemma} |
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413 |
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414 \begin{proof} |
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415 Apply Lemma \ref {xxyy3} repeatedly. |
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416 \end{proof} |
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417 |
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418 \begin{lemma} \label{xxyy5} |
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419 Let $M$ be a Riemannian $n$-manifold and positive integers $m$ and $k$. |
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420 There exists a constant $\eta(M, m, k)$ such that for all subsets |
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421 $S\subeq M$ which are contained in a (not necessarily disjoint) union of |
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422 $k$ metric balls of radius $r$, $r < \eta(M, m, k)$, |
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423 there exist neighborhoods $U_0, U_1, \ldots, U_m$ of $S$, |
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424 each homeomorphic to a disjoint union of balls, such that |
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425 \[ |
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426 \Nbd_{2r}(S) \subeq U_0 \subeq \Nbd_{4^k r}(S) |
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427 \subeq U_1 \subeq \Nbd_{4^{2k} r}(S) \cdots |
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428 \subeq U_m \subeq \Nbd_{4^{(m+1)k} r}(S) . |
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429 \] |
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430 |
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431 \end{lemma} |
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432 |
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433 \begin{proof} |
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434 Choose $\eta = \eta(M, m, k)$ small enough so that metric balls of radius $4^{(m+1)k} \eta$ |
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435 are injective and also have small distortion with respect to a Euclidean metric. |
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436 Then proceed as in the proof of Lemma \ref{xxyy4}. |
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437 \end{proof} |
352 |
438 |
353 \medskip |
439 \medskip |
354 |
440 |
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441 |
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442 |
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443 |
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444 |
355 \noop{ |
445 \noop{ |
356 |
446 |
357 \begin{lemma} |
447 \begin{lemma} |
358 |
448 |
359 \end{lemma} |
449 \end{lemma} |
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450 |
360 \begin{proof} |
451 \begin{proof} |
361 |
452 |
362 \end{proof} |
453 \end{proof} |
363 |
454 |
364 } |
455 } |