I'm trying to understand the proof of the Lemma 1.3.1 from K-theory of Atiyah Notes from Benjamin, but there are so parts which I can't, so I'll be really great full for any help.
Notation: $E,F$ denote vector bundles.
Lemma 1.2.1. If $\varphi: F \rightarrow E$ is a monomorphism, then $\varphi(F)$ is a sub-bundle of $E,$ and $\varphi: F \rightarrow \varphi(F)$ is an isomorphism.
Proof: $\varphi: F \rightarrow \varphi(F)$ is a bijection, so if $\varphi(F)$ is a sub-bundle, $\varphi: F \rightarrow \varphi(F)$ is an isomorphism. Thus we need only show that $\varphi: F \rightarrow \varphi(F)$ is an isomorphism.
The problem is local, so it suffices to consider the case when $E$ and $F$ are product bundles. Let $E = X \times V$ (I suppose $V$ is a $n-$dimensional $\mathbb{C}-$vector space ) and let $x \in X;$ choose $W_{x} \subset V$ to be a subspace complementary to $\varphi(F)$ (i.e $\varphi(F) \oplus W_{x}=V $ and $\varphi(F) \cap W_{x} ={0}.$ ) Then $G=X \times W_{x}$ is a sub-bundle of E.
Define $\theta: F \oplus G \rightarrow E$ by $\theta(a \oplus b)= \varphi(a)+i(b),$ where $i:G \rightarrow E$ is the inclusion. By construction $\theta_{x}$ is an isomorphism. Thus there exists an open neighborhood $U$ of x such that $\theta|U$ is an isomorphism. F is a sub-bundle of $ F \oplus G,$ so $\theta(F)=\varphi(F)$ is a sub-bundle of $\theta(F \oplus G)= E$ on $U.$
Question How can I show that $G=X \times W_{x}$ is a sub-bundle of E?