I am referring to theorem 9.24 of Dan Freed's notes. The theorem establishes space of Fredholm operators as representing space of K-theory i.e. $$\text{ind}: [X,\text{Fred}(H)] \to K(X) $$ is an isomorphism of abelian groups.
In proving the surjectivity of the above map the notes make the following statements: any element of $K(X)$ has the form $[E] - N$ for some vector bundle $E \to X$ and $N \in \mathbb{Z}^{\geq 0}$, embed $E$ in a trivial bundle $X \times \mathbb{E}$ and define $p_x \in \text{End}(X \times \mathbb{E})$ as orthogonal projection with kernel $E_x$, finally embed $\mathbb{E}$ into $H$ and extend $p_x$ to be the identity on $\mathbb{E}^{\perp}$.
I am quite confused by this brief proof, in particular is the orthogonal projection $p_x$ the Fredholm operator? If so what is the subspace $W$ of $\mathbb{E}$ that is transverse to $p_x$ for any $x$ (transverse means $p_x(\mathbb{E}) + W = \mathbb{E}$, ref. pp.3 of the same notes)? Note that the ind map above is built on the map on the space of Fredholm operators $T \mapsto [T^*{K_W}] - [X \times W]$ (eqn 9.23, definition of the pullback bundle $T^*{K_W} \to X$ is given in Lemma 9.16 of the same notes) so the definition of the left-hand side of the ind map is determined by the specification of the Fredholm operator $T$ and the transverse subspace $W$.