Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.

For a continuous map $Tcolon Xto F$, there exists a closed subspace $Wsubseteq H$ with $dim H/W<infty$ such that $Wcapker T_x=0$ for all $xin X$ and $H/T(W) =bigcup_{xin X} H/T_x(W)$ is a vector bundle over $X$ (See appendix of K-Theory, Anderson & Atiyah).
Then one can show that
$$mbox{Ind}_1(T) = [Xtimes H/W] – [H/T(W)] in K(X)$$
does not depend on $W$.

On the other hand, there exists a finite dimensional subspace $Vsubseteq H$ such that $V+T_x(H) = H$ for all $xin X$, so we can define $T^Vcolon Xto F(Hoplus V, H)$ by $T^V_x(u,v) = T_x u + v$. Then $T^V_x$ is surjective and $dimker T^V_x$ is constant on $x$. Thus $ker T^V = bigcup_{xin X} ker T_x$ is also a vector bundle over $X$. One can show that
$$ mbox{Ind}_2(T) = [ker T^V] – [Xtimes V] in K(X)$$
does not depend on $V$.

These index maps are called the family index of families of Fredholm operators in $H$, and it made me suspect that they are equal.

Question: Is it true that $$[Xtimes H/W] – [H/T(W)] = [ker T^V] – [Xtimes V] qquad (1)$$ in $K(X)$ ?
Is there any reference that proves the equivalence of these indexes?

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Edit: We can shrink $W$ or augment $V$ in order to have $dim H/W = dim V$. Say $H/W cong V cong mathbb{C}^N$, so that $Xtimes H/W cong Xtimes V cong Xtimesmathbb{C}^N$, and therefore
$$mbox{Ind}_1(T) = [Xtimesmathbb{C}^N] – [H/T(W)] ,$$
$$mbox{Ind}_2(T) = [ker T^V] – [Xtimesmathbb{C}^N] .$$

Equation $(1)$ becomes
$$[Xtimesmathbb{C}^N] – [H/T(W)] = [ker T^V] – [Xtimesmathbb{C}^N]$$
and it holds iff there exists $kgeq0$ such that

$$ker T^V oplus H/T(W) oplus (Xtimesmathbb{C}^k) cong Xtimesmathbb{C}^{2N+k}$$
But why does there exists such $k$?