[proofplan] We construct the projection on the quotient by using the original vector bundle projections on the two pieces and check that it is well-defined because $\varphi$ lies over the identity on $C$. Away from the overlap $C$, the vector bundle charts are inherited directly from $E_A$ and $E_B$. Near $C$, the collar coordinates and the boundary matrix $g:C\to GL(k,\mathbb R)$ allow us to rewrite the $B$-side fibre coordinate by $g(x)^{-1}$, producing one product chart over the glued collar. The collar chart is taken as part of the defining local atlas near $C$; compatibility with the inherited charts gives the required $C^r$ vector bundle structure without requiring a separate jet-matching argument for a piecewise inverse. Since $X$ is already part of the closed bundle-gluing cover in the relevant category, the Hausdorff and smooth-base requirements are treated as hypotheses on the base rather than reproved from the quotient presentation. The natural maps from the original bundles identify the pullback restrictions to $A$ and $B$ with the prescribed boundary gluing. [/proofplan]

[step:Define the quotient projection and the fibrewise vector space structure] Let $q:E_A\sqcup E_B\to E$ be the quotient map. Define $\pi:E\to X$ by setting $\pi(q(e))=\pi_A(e)$ for $e\in E_A$ and $\pi(q(e))=\pi_B(e)$ for $e\in E_B$. This is well-defined: if $e_A\in E_A|_C$ and $e_B=\varphi(e_A)$, then $\pi_B(e_B)=\pi_A(e_A)$ because $\varphi$ is a vector bundle isomorphism over $\operatorname{id}_C$. We use the stated closed bundle-gluing cover hypothesis together with the explicit compatible-collar assumptions as follows: $X$ is already a Hausdorff [topological space](/page/Topological%20Space), respectively a smooth space with the declared glued collar atlas near $C$. Thus the collar neighbourhood used below is an actual neighbourhood in the base category, not merely a set-theoretic pushout. For each $x\in X$, the fibre $E_x=\pi^{-1}(\{x\})$ is defined as follows. If $x\in A\setminus C$, then $E_x$ is identified with $(E_A)_x$ through $q$. If $x\in B\setminus C$, then $E_x$ is identified with $(E_B)_x$ through $q$. If $x\in C$, then \begin{align*} E_x=((E_A)_x\sqcup (E_B)_x)/\{e_A\sim \varphi(e_A)\}, \end{align*} and the map $(E_A)_x\to E_x$, $e_A\mapsto q(e_A)$, is a linear isomorphism because $\varphi_x:(E_A)_x\to (E_B)_x$ is a linear isomorphism. We transfer the [vector space](/page/Vector%20Space) operations from $(E_A)_x$ to $E_x$ through this isomorphism. This definition agrees with the vector space operations inherited from $(E_B)_x$, since $\varphi_x$ is linear. [/step]

[step:Use the original bundle charts away from the gluing locus] Let $x\in A\setminus C$. Since $A\setminus C$ is open in $X$ near $x$, choose an open neighbourhood $U\subset A\setminus C$ of $x$ and a $C^r$ vector bundle trivialization \begin{align*} \theta_A:E_A|_U\to U\times \mathbb R^k. \end{align*} The map \begin{align*} \Theta_A:E|_U\to U\times \mathbb R^k \end{align*} defined by \begin{align*} \Theta_A(q(e))=\theta_A(e),\qquad e\in E_A|_U, \end{align*} is well-defined because no gluing identifications occur over $U\subset A\setminus C$. It is a fibrewise linear bijection and is a homeomorphism, respectively a diffeomorphism, because it is induced from the original bundle chart on $E_A$. Let $x\in B\setminus C$. Since $B\setminus C$ is open in $X$ near $x$, choose an open neighbourhood $V\subset B\setminus C$ of $x$ and a $C^r$ vector bundle trivialization \begin{align*} \theta_B:E_B|_V\to V\times \mathbb R^k. \end{align*} Define \begin{align*} \Theta_B:E|_V\to V\times \mathbb R^k \end{align*} by \begin{align*} \Theta_B(q(e))=\theta_B(e),\qquad e\in E_B|_V. \end{align*} This is well-defined because no gluing identifications occur over $V\subset B\setminus C$. It is a fibrewise linear bijection and is a homeomorphism, respectively a diffeomorphism, because it is induced from the original bundle chart on $E_B$. [/step]

[step:Build one collar trivialization across the glued boundary]By the compatible-collar hypothesis, choose $\varepsilon>0$ and collar maps $\kappa_A:C\times[0,\varepsilon)\to A$ and $\kappa_B:C\times[0,\varepsilon)\to B$ such that the induced neighbourhood $U_C\subset X$ of $C$ is identified with $C\times(-\varepsilon,\varepsilon)$ by sending $\kappa_A(x,s)$ to $(x,-s)$ and $\kappa_B(x,s)$ to $(x,s)$. We use the given collar trivializations $\tau_A:E_A|_{\kappa_A(C\times[0,\varepsilon))}\to \kappa_A(C\times[0,\varepsilon))\times\mathbb R^k$ and $\tau_B:E_B|_{\kappa_B(C\times[0,\varepsilon))}\to \kappa_B(C\times[0,\varepsilon))\times\mathbb R^k$. Define the map $\Theta_C:E|_{U_C}\to U_C\times\mathbb R^k$ as follows. If $e_A\in E_A$ lies over $\kappa_A(x,s)$ with $s\in[0,\varepsilon)$ and \begin{align*} \tau_A(e_A)=(\kappa_A(x,s),v), \end{align*} set \begin{align*} \Theta_C(q(e_A))=((x,-s),v). \end{align*} If $e_B\in E_B$ lies over $\kappa_B(x,s)$ with $s\in[0,\varepsilon)$ and \begin{align*} \tau_B(e_B)=(\kappa_B(x,s),w), \end{align*} set \begin{align*} \Theta_C(q(e_B))=((x,s),g(x)^{-1}w). \end{align*} This definition is compatible with the quotient relation over $C$. Indeed, if $e_A\in (E_A)_x$ and $\tau_A(e_A)=(x,v)$, then \begin{align*} \tau_B(\varphi(e_A))=(x,g(x)v), \end{align*} so the $B$-side formula gives \begin{align*} \Theta_C(q(\varphi(e_A)))=((x,0),g(x)^{-1}g(x)v)=((x,0),v)=\Theta_C(q(e_A)). \end{align*} Thus $\Theta_C$ is well-defined. It is fibrewise linear by construction. It is bijective because, over $t<0$, its inverse is obtained from $\tau_A^{-1}$, over $t>0$, its inverse is obtained from $\tau_B^{-1}$ after multiplying the fibre coordinate by $g(x)$, and over $t=0$ either side gives the same quotient point. We first prove that $\Theta_C$ is a homeomorphism in the topological category. Let $W_A\subset E_A$ be the preimage of $\kappa_A(C\times[0,\varepsilon))$ under $\pi_A$, and let $W_B\subset E_B$ be the preimage of $\kappa_B(C\times[0,\varepsilon))$ under $\pi_B$. The collar images are open neighbourhoods of $C$ in the relative topologies of $A$ and $B$, so $W_A\sqcup W_B$ is open in $E_A\sqcup E_B$. It is also saturated for the quotient relation, because the only identifications occur over $C$ and both collar preimages contain the fibres over $C$. Hence the restriction \begin{align*} q_C:W_A\sqcup W_B\to q(W_A\sqcup W_B)=E|_{U_C} \end{align*} is a quotient map: if $O\subset E|_{U_C}$ and $q_C^{-1}(O)$ is open in $W_A\sqcup W_B$, then $q_C^{-1}(O)=(W_A\sqcup W_B)\cap G$ for some open $G\subset E_A\sqcup E_B$, and the saturated set \begin{align*} G\cup ((E_A\sqcup E_B)\setminus (W_A\sqcup W_B)) \end{align*} has quotient image whose intersection with $E|_{U_C}$ is $O$. Define a map \begin{align*} \widetilde{\Theta}_C:W_A\sqcup W_B\to U_C\times\mathbb R^k \end{align*} by the same two formulas used for $\Theta_C$. On $W_A$ it is the composition of $\tau_A$ with the continuous collar-coordinate map $(\kappa_A(x,s),v)\mapsto ((x,-s),v)$. On $W_B$ it is the composition of $\tau_B$ with the continuous map $(\kappa_B(x,s),w)\mapsto ((x,s),g(x)^{-1}w)$. The inversion map $GL(k,\mathbb R)\to GL(k,\mathbb R)$ and matrix-vector multiplication are continuous, so the $B$-side formula is continuous. The compatibility computation above shows that $\widetilde{\Theta}_C$ is constant on the equivalence classes defining $E|_{U_C}$. Therefore, by the universal property of quotient maps applied to the quotient map $q_C$, the induced map $\Theta_C:E|_{U_C}\to U_C\times\mathbb R^k$ is continuous. To justify continuity of the inverse in the topological category, define $\Psi_C:U_C\times\mathbb R^k\to E|_{U_C}$ by using $q\circ\tau_A^{-1}$ on \begin{align*} P_A=C\times(-\varepsilon,0]\times\mathbb R^k \end{align*} with $s=-t$, and by using $q\circ\tau_B^{-1}$ after replacing the fibre coordinate $u$ by $g(x)u$ on \begin{align*} P_B=C\times[0,\varepsilon)\times\mathbb R^k \end{align*} with $s=t$. These are subsets of $U_C\times\mathbb R^k=C\times(-\varepsilon,\varepsilon)\times\mathbb R^k$. The interval $(-\varepsilon,0]$ is closed in $(-\varepsilon,\varepsilon)$ because it is $(-\varepsilon,\varepsilon)\cap(-\infty,0]$, and $[0,\varepsilon)$ is closed in $(-\varepsilon,\varepsilon)$ because it is $(-\varepsilon,\varepsilon)\cap[0,\infty)$. Therefore $P_A$ and $P_B$ are closed in $U_C\times\mathbb R^k$, and they cover $U_C\times\mathbb R^k$. Each restriction is continuous because it is a composition of continuous maps, and the two restrictions agree on $P_A\cap P_B=C\times\{0\}\times\mathbb R^k$ because $\tau_B(\varphi(e_A))=(x,g(x)v)$ whenever $\tau_A(e_A)=(x,v)$. By the [Gluing Lemma](/theorems/1871) for maps defined on closed subsets, $\Psi_C$ is continuous. The displayed formulas show $\Psi_C\circ\Theta_C=\operatorname{id}_{E|_{U_C}}$ and $\Theta_C\circ\Psi_C=\operatorname{id}_{U_C\times\mathbb R^k}$, so $\Theta_C$ is a homeomorphism in the topological category. In the smooth category, we use this already established homeomorphism to define the local smooth structure on the same quotient topological space $E|_{U_C}$ by transporting the product smooth structure through $\Theta_C$. Since the topology transported by $\Theta_C$ is exactly the quotient [subspace topology](/page/Subspace%20Topology) just proved, this does not change the underlying topological space. Thus $\Theta_C$ is a diffeomorphism by definition of this local smooth chart; the next step verifies that this chart is smoothly compatible with the inherited bundle charts from $E_A$ and $E_B$. No conclusion about smoothness is being drawn merely from agreement of the two inverse formulas at $t=0$.[/step]

[guided]The only delicate point in the construction is the chart near $C$, because points over $C$ have two representatives before taking the quotient. The base collar identifies a neighbourhood of $C$ in $X$ with $C\times(-\varepsilon,\varepsilon)$: the $A$-side is represented by negative collar coordinate $t=-s$, and the $B$-side by positive collar coordinate $t=s$. In the chosen trivializations, an element over the $A$-side has coordinates $(\kappa_A(x,s),v)$, while an element over the $B$-side has coordinates $(\kappa_B(x,s),w)$. On the boundary $s=0$, the gluing rule is not $v=w$; it is \begin{align*} w=g(x)v. \end{align*} Therefore a single product coordinate across the glued collar must correct the $B$-side coordinate. We do this by replacing $w$ with $g(x)^{-1}w$. Define $\Theta_C:E|_{U_C}\to U_C\times\mathbb R^k$ by setting $\Theta_C(q(e_A))=((x,-s),v)$ when $\tau_A(e_A)=(\kappa_A(x,s),v)$, and by setting $\Theta_C(q(e_B))=((x,s),g(x)^{-1}w)$ when $\tau_B(e_B)=(\kappa_B(x,s),w)$. We now check the boundary compatibility, which is the point of inserting $g(x)^{-1}$. If $e_A\in (E_A)_x$ and $\tau_A(e_A)=(x,v)$, then the hypothesis on $\varphi$ gives \begin{align*} \tau_B(\varphi(e_A))=(x,g(x)v). \end{align*} Thus \begin{align*} \Theta_C(q(\varphi(e_A)))=((x,0),g(x)^{-1}g(x)v)=((x,0),v)=\Theta_C(q(e_A)). \end{align*} So the formula descends to the quotient. The map is fibrewise linear because each formula is linear in the fibre coordinate and $g(x)^{-1}:\mathbb R^k\to\mathbb R^k$ is a linear isomorphism for each $x\in C$. It is bijective because for $t<0$ we recover the unique $A$-side representative using $\tau_A^{-1}$, for $t>0$ we recover the unique $B$-side representative using $\tau_B^{-1}$ after replacing a coordinate $u\in\mathbb R^k$ by $g(x)u$, and for $t=0$ the two possible representatives are identified by the quotient relation. Now separate the topological and smooth points. In the topological category, we must prove continuity of $\Theta_C$ as a map out of a quotient space, not only continuity of its inverse. Let $W_A\subset E_A$ be the preimage of the $A$-collar under $\pi_A$, and let $W_B\subset E_B$ be the preimage of the $B$-collar under $\pi_B$. The set $W_A\sqcup W_B$ is open in $E_A\sqcup E_B$ because collar neighbourhoods of the boundary are open in the relative topologies of $A$ and $B$. It is saturated for the quotient relation because every point identified with a point in one collar fibre lies in the corresponding collar fibre on the other side. Therefore the restricted map $q_C:W_A\sqcup W_B\to E|_{U_C}$ is itself a quotient map: this is the standard restriction property for quotient maps on open saturated subsets, and it applies here exactly because saturation prevents an equivalence class from being cut in half. Define $\widetilde{\Theta}_C:W_A\sqcup W_B\to U_C\times\mathbb R^k$ by the same formulas as $\Theta_C$. The $A$-side formula is continuous because it is obtained from $\tau_A$ and the collar coordinate change $s\mapsto -s$. The $B$-side formula is continuous because it is obtained from $\tau_B$, the continuous map $g:C\to GL(k,\mathbb R)$, the continuous inversion map on $GL(k,\mathbb R)$, and continuous matrix-vector multiplication. The boundary computation already made shows that $\widetilde{\Theta}_C$ has the same value on representatives identified by the quotient relation. Hence the universal property of quotient maps, applied to the quotient map $q_C$, gives a continuous induced map $\Theta_C:E|_{U_C}\to U_C\times\mathbb R^k$. Next define the inverse candidate $\Psi_C:U_C\times\mathbb R^k\to E|_{U_C}$ by those same inverse formulas. Its restriction to $P_A=C\times(-\varepsilon,0]\times\mathbb R^k$ is the continuous map $q\circ\tau_A^{-1}$, with $s=-t$. Its restriction to $P_B=C\times[0,\varepsilon)\times\mathbb R^k$ is the continuous map obtained by first sending $(x,t,u)$ to $(x,t,g(x)u)$ and then applying $q\circ\tau_B^{-1}$, with $s=t$. These sets are subsets of $U_C\times\mathbb R^k=C\times(-\varepsilon,\varepsilon)\times\mathbb R^k$. Since $(-\varepsilon,0]=(-\varepsilon,\varepsilon)\cap(-\infty,0]$ and $[0,\varepsilon)=(-\varepsilon,\varepsilon)\cap[0,\infty)$, both intervals are closed in the collar interval $(-\varepsilon,\varepsilon)$. Hence $P_A$ and $P_B$ are closed in $U_C\times\mathbb R^k$, and they cover all of $U_C\times\mathbb R^k$. The two formulas agree on $P_A\cap P_B=C\times\{0\}\times\mathbb R^k$ because $\varphi$ sends the $A$-coordinate $v$ to the $B$-coordinate $g(x)v$. The [Gluing Lemma](/theorems/1871) therefore gives continuity of $\Psi_C$. Since $\Psi_C$ and $\Theta_C$ compose to the identity maps on both sides, $\Theta_C$ is a homeomorphism. In the smooth category, we do not infer smoothness of $\Psi_C$ merely from agreement of its values at $t=0$. Instead, we first use the homeomorphism just proved to know that the [quotient topology](/page/Quotient%20Topology) on $E|_{U_C}$ is exactly the topology transported from $U_C\times\mathbb R^k$ by $\Theta_C$. We then define the smooth structure near $C$ by this local coordinate chart on that same underlying topological space. With that definition, $\Theta_C$ is a smooth product trivialization by construction, and smooth compatibility with the original charts is checked in the transition-function step below.[/guided]

[step:Verify that the transition functions are in the chosen category] The charts constructed away from $C$ are original $C^r$ vector bundle charts for $E_A$ or $E_B$. Their mutual transition functions are therefore $C^r$ maps into $GL(k,\mathbb R)$. It remains to compare an original chart on one side with the collar chart $\Theta_C$. On the $A$-side, the transition from $\Theta_C$ to any $A$-bundle chart is the transition between two $C^r$ trivializations of $E_A$. Hence it is a $C^r$ map into $GL(k,\mathbb R)$. On the $B$-side, the transition differs from the transition between two $C^r$ trivializations of $E_B$ by composition with the fibre automorphism \begin{align*} C\times[0,\varepsilon)\times\mathbb R^k\to C\times[0,\varepsilon)\times\mathbb R^k \end{align*} sending $(x,s,v)$ to $(x,s,g(x)v)$. This map is $C^r$ because it is the product-collar pullback of the $C^r$ map $g:C\to GL(k,\mathbb R)$ followed by smooth matrix-vector multiplication. Thus this transition is again $C^r$ and fibrewise linear. Therefore all transition maps for the constructed atlas are $C^r$ vector bundle transition functions. [/step]

[step:Identify the restrictions with the original bundles] Throughout this step, $E|_A$ and $E|_B$ mean the pullback restrictions along the inclusions $A\hookrightarrow X$ and $B\hookrightarrow X$; equivalently, their fibres over $x\in A$ or $x\in B$ are the quotient fibres $E_x$, not a pre-quotient subset of $E_A\sqcup E_B$. Define $\iota_A:E_A\to E|_A$ by $\iota_A(e_A)=q(e_A)$, and define $\iota_B:E_B\to E|_B$ by $\iota_B(e_B)=q(e_B)$. Over $A\setminus C$ and $B\setminus C$, these maps are the evident identifications. Over $C$, the map $\iota_A$ is a fibrewise linear isomorphism from $(E_A)_x$ to the quotient fibre $E_x$, and $\iota_B$ is a fibrewise linear isomorphism from $(E_B)_x$ to the same quotient fibre $E_x$. They satisfy \begin{align*} \iota_B(\varphi(e_A))=q(\varphi(e_A))=q(e_A)=\iota_A(e_A) \end{align*} for every $e_A\in E_A|_C$. In the charts away from $C$, $\iota_A$ and $\iota_B$ are the original bundle chart identifications. In the collar chart $\Theta_C$, the map $\iota_A$ is represented by $(x,s,v)\mapsto (x,-s,v)$ on the $A$-collar, and $\iota_B$ is represented by $(x,s,w)\mapsto (x,s,g(x)^{-1}w)$ on the $B$-collar. These coordinate representations are homeomorphisms, respectively diffeomorphisms, and are fibrewise linear. Thus $\iota_A$ and $\iota_B$ are vector bundle isomorphisms onto the restrictions $E|_A$ and $E|_B$, and the two restrictions recover $E_A$ and $E_B$ while agreeing over $C$ through the prescribed identification $\varphi$. The preceding steps give a rank $k$ real $C^r$ vector bundle structure on $\pi:E\to X$, with the stated restriction property. This completes the proof. [/step]