[proofplan]
We prove the homotopy statement by converting a homotopy of clutching functions into a single vector bundle over the cylinder $M\times[0,1]$. The endpoint restrictions of this cylinder bundle are exactly the two clutched bundles $E_{\gamma_0}$ and $E_{\gamma_1}$. A standard interval-homotopy invariance result for smooth vector bundles, obtained by parallel transport in the interval direction, identifies these endpoint restrictions. Finally, the frame-change formula is checked directly from the defining [equivalence relation](/page/Equivalence%20Relation).
[/proofplan]
[step:Set up the clutched bundle notation using the fixed gluing data]
Let $i_A:C\to A$ and $i_B:C\to B$ be the smooth embeddings identifying the common gluing locus in the two pieces, as in the theorem statement. The quotient manifold is
\begin{align*}
M=A\cup_C B,
\end{align*}
where points $i_A(c)\in A$ and $i_B(c)\in B$ are identified for every $c\in C$.
For any smooth clutching function $\eta:C\to GL(k,\mathbb R)$, denote by $E_\eta\to M$ the rank-$k$ vector bundle obtained by gluing the product bundles $A\times\mathbb R^k\to A$ and $B\times\mathbb R^k\to B$ through the relation
\begin{align*}
(i_B(c),v)_B \sim (i_A(c),\eta(c)v)_A
\end{align*}
for $c\in C$ and $v\in\mathbb R^k$. The quotient smooth structure on $M$ and the smooth gluing data ensure that this relation defines the usual smooth vector bundle obtained from the transition function $\eta$ on the overlap represented by $C$.
[/step]
[step:Build a vector bundle over the cylinder from the clutching homotopy]
Because $\gamma_0$ and $\gamma_1$ are smoothly homotopic, fix a smooth map
\begin{align*}
\Gamma:C\times[0,1]\to GL(k,\mathbb R)
\end{align*}
such that $\Gamma(c,0)=\gamma_0(c)$ and $\Gamma(c,1)=\gamma_1(c)$ for every $c\in C$.
Taking the product of the stated smooth gluing data with $[0,1]$ gives the smooth gluing model
\begin{align*}
M\times[0,1]=(A\times[0,1])\cup_{C\times[0,1]}(B\times[0,1]),
\end{align*}
where $(i_A(c),t)\in A\times[0,1]$ is identified with $(i_B(c),t)\in B\times[0,1]$ for every $(c,t)\in C\times[0,1]$.
Let $\pi_A$ be the product rank-$k$ bundle
\begin{align*}
\pi_A:(A\times[0,1])\times\mathbb R^k \to A\times[0,1].
\end{align*}
Let $\pi_B$ be the product rank-$k$ bundle
\begin{align*}
\pi_B:(B\times[0,1])\times\mathbb R^k \to B\times[0,1].
\end{align*}
Define a smooth rank-$k$ vector bundle
\begin{align*}
\mathcal E_\Gamma \to M\times[0,1]
\end{align*}
by gluing these two product bundles along $C\times[0,1]$ through the equivalence relation
\begin{align*}
(i_B(c),t,v)_B \sim (i_A(c),t,\Gamma(c,t)v)_A
\end{align*}
for every $(c,t)\in C\times[0,1]$ and every $v\in\mathbb R^k$.
The clutching construction is valid in this gluing model because the overlap on which the transition function is prescribed is exactly $C\times[0,1]$, and the map
\begin{align*}
\Gamma:C\times[0,1]\to GL(k,\mathbb R)
\end{align*}
is smooth with values in invertible linear maps. Hence it supplies a smooth $GL(k,\mathbb R)$-valued transition function for the product gluing cover, so the quotient inherits the standard smooth vector bundle structure with the two displayed product bundles as local trivializations.
[/step]
[step:Identify the endpoint restrictions with the two clutched bundles]
For $r\in[0,1]$, let $j_r$ be the smooth inclusion of the $r$-th slice, defined by
\begin{align*}
j_r:M\to M\times[0,1],\qquad j_r(x)=(x,r).
\end{align*} The pullback bundle $j_r^*\mathcal E_\Gamma\to M$ is obtained by restricting the preceding clutching construction to $A\times\{r\}$ and $B\times\{r\}$. Therefore its clutching function is the smooth map $\gamma_r$ defined by
\begin{align*}
\gamma_r:C\to GL(k,\mathbb R),\qquad \gamma_r(c)=\Gamma(c,r).
\end{align*}
In particular,
\begin{align*}
j_0^*\mathcal E_\Gamma \cong E_{\gamma_0}.
\end{align*}
Also,
\begin{align*}
j_1^*\mathcal E_\Gamma \cong E_{\gamma_1}.
\end{align*}
[/step]
[step:Use parallel transport in the interval direction to identify the endpoint bundles]
Since $M$ is a smooth paracompact manifold by hypothesis, $M\times[0,1]$ is also a smooth paracompact manifold. The smooth bundle $\mathcal E_\Gamma\to M\times[0,1]$ therefore admits a smooth connection by the standard partition-of-unity construction on a locally finite trivializing cover. Choose such a connection on $\mathcal E_\Gamma$.
For each $x\in M$, define the smooth vertical path
\begin{align*}
\sigma_x:[0,1]\to M\times[0,1],\qquad \sigma_x(t)=(x,t).
\end{align*}
Parallel transport for the chosen connection along $\sigma_x$ gives a linear isomorphism $(\mathcal E_\Gamma)_{(x,0)}\to(\mathcal E_\Gamma)_{(x,1)}$. Smooth dependence of ordinary differential equation solutions on parameters implies that these fiberwise maps vary smoothly with $x$, so they assemble into a smooth vector bundle isomorphism
\begin{align*}
j_0^*\mathcal E_\Gamma \cong j_1^*\mathcal E_\Gamma.
\end{align*}
Thus we have an isomorphism
\begin{align*}
j_0^*\mathcal E_\Gamma \cong j_1^*\mathcal E_\Gamma.
\end{align*}
Using the endpoint identifications from the previous step, we obtain
\begin{align*}
E_{\gamma_0}\cong E_{\gamma_1}.
\end{align*}
[guided]
We spell out the endpoint-invariance argument from the beginning so that this guided step can be read independently. Let
\begin{align*}
\Gamma:C\times[0,1]\to GL(k,\mathbb R)
\end{align*}
be the smooth homotopy with $\Gamma(c,0)=\gamma_0(c)$ and $\Gamma(c,1)=\gamma_1(c)$ for every $c\in C$. Glue the product bundles $(A\times[0,1])\times\mathbb R^k\to A\times[0,1]$ and $(B\times[0,1])\times\mathbb R^k\to B\times[0,1]$ along $C\times[0,1]$ by
\begin{align*}
(c,t,v)_B \sim (c,t,\Gamma(c,t)v)_A.
\end{align*}
Because $\Gamma$ is smooth and takes values in invertible linear maps, this is a smooth $GL(k,\mathbb R)$-valued transition function, and therefore defines a smooth rank-$k$ vector bundle
\begin{align*}
\mathcal E_\Gamma\to M\times[0,1].
\end{align*}
For $r\in[0,1]$, define the slice inclusion
\begin{align*}
j_r:M\to M\times[0,1],\qquad j_r(x)=(x,r).
\end{align*}
Restricting the preceding gluing relation to the slice $t=r$ gives the clutching relation
\begin{align*}
(c,v)_B \sim (c,\Gamma(c,r)v)_A.
\end{align*}
Thus $j_r^*\mathcal E_\Gamma$ is the clutched bundle $E_{\gamma_r}$, where $\gamma_r:C\to GL(k,\mathbb R)$ is the smooth map $\gamma_r(c)=\Gamma(c,r)$. In particular,
\begin{align*}
j_0^*\mathcal E_\Gamma\cong E_{\gamma_0},\qquad j_1^*\mathcal E_\Gamma\cong E_{\gamma_1}.
\end{align*}
We now prove the endpoint identification directly by parallel transport. The hypothesis that $M$ is a smooth paracompact manifold matters here because it implies that $M\times[0,1]$ is also paracompact, and paracompactness gives smooth partitions of unity subordinate to locally finite trivializing covers. Using such a [partition of unity](/page/Partition%20of%20Unity), choose a smooth connection on the vector bundle $\mathcal E_\Gamma\to M\times[0,1]$.
For each $x\in M$, define the vertical path
\begin{align*}
\sigma_x:[0,1]\to M\times[0,1],\qquad \sigma_x(t)=(x,t).
\end{align*}
Parallel transport along $\sigma_x$ gives a linear isomorphism from the fiber $(\mathcal E_\Gamma)_{(x,0)}$ to the fiber $(\mathcal E_\Gamma)_{(x,1)}$. In local trivializations, parallel transport is determined by a linear ordinary differential equation whose coefficients are smooth in both $x$ and $t$; the smooth dependence theorem for ordinary differential equations therefore implies that the resulting linear maps depend smoothly on $x$. Hence the fiberwise parallel-transport maps assemble into a smooth vector bundle isomorphism
\begin{align*}
j_0^*\mathcal E_\Gamma \cong j_1^*\mathcal E_\Gamma.
\end{align*}
Combining this isomorphism with the endpoint identifications above yields
\begin{align*}
E_{\gamma_0}\cong E_{\gamma_1}.
\end{align*}
[/guided]
[/step]
[step:Compute how changing the two frames transforms the clutching function]
Let $a:A\to GL(k,\mathbb R)$ and $b:B\to GL(k,\mathbb R)$ be smooth maps, and define
\begin{align*}
\gamma':C\to GL(k,\mathbb R),\qquad \gamma'(c)=a(c)\gamma(c)b(c)^{-1}.
\end{align*}
Define a map on representatives by
\begin{align*}
\Phi_A:A\times\mathbb R^k\to A\times\mathbb R^k,\qquad \Phi_A(x,v)=(x,a(x)v),
\end{align*}
and
\begin{align*}
\Phi_B:B\times\mathbb R^k\to B\times\mathbb R^k,\qquad \Phi_B(x,v)=(x,b(x)v).
\end{align*}
We check that these two maps descend to a well-defined bundle map
\begin{align*}
\Phi:E_\gamma &\to E_{\gamma'}.
\end{align*}
In $E_\gamma$, the gluing relation is
\begin{align*}
(i_B(c),v)_B \sim (i_A(c),\gamma(c)v)_A.
\end{align*}
Applying $\Phi_B$ to the left representative and $\Phi_A$ to the right representative gives
\begin{align*}
(i_B(c),b(i_B(c))v)_B
\end{align*}
and
\begin{align*}
(i_A(c),a(i_A(c))\gamma(c)v)_A.
\end{align*}
In $E_{\gamma'}$, the gluing relation gives
\begin{align*}
(i_B(c),b(i_B(c))v)_B \sim (i_A(c),\gamma'(c)b(i_B(c))v)_A.
\end{align*}
Since $b(c)^{-1}b(c)$ is the identity [linear map](/page/Linear%20Map) on $\mathbb R^k$, we have
\begin{align*}
\gamma'(c)b(i_B(c))v=a(i_A(c))\gamma(c)b(i_B(c))^{-1}b(i_B(c))v=a(i_A(c))\gamma(c)v,
\end{align*}
so the two images are equivalent in $E_{\gamma'}$. Thus $\Phi$ is well-defined.
The inverse is induced on the $A$-piece by
\begin{align*}
A\times\mathbb R^k\to A\times\mathbb R^k,\qquad (x,v)\mapsto (x,a(x)^{-1}v),
\end{align*}
and on the $B$-piece by
\begin{align*}
B\times\mathbb R^k\to B\times\mathbb R^k,\qquad (x,v)\mapsto (x,b(x)^{-1}v).
\end{align*}
Therefore $\Phi$ is a smooth vector bundle isomorphism over $M$, and hence
\begin{align*}
E_\gamma\cong E_{a\gamma b^{-1}}.
\end{align*}
[/step]
[step:Conclude both invariance statements]
The homotopy argument proves that smoothly homotopic clutching functions define isomorphic rank-$k$ vector bundles. The direct frame-change computation proves that replacing the trivializing frame on $A$ by $a$ and the trivializing frame on $B$ by $b$ replaces the clutching function by
\begin{align*}
c\mapsto a(c)\gamma(c)b(c)^{-1}
\end{align*}
without changing the isomorphism class of the resulting bundle. These are exactly the two assertions of the theorem.
[/step]
