Homology braid I (Ex)

Consider the following commutative braid of exact sequences:

$\displaystyle \xymatrix{ A \ar[dr] \ar@/^2pc/[rr]^{\alpha} && B \ar[dr] \ar@/^2pc/[rr]^{\beta} && C \\ & D \ar[dr] \ar[ur] && E \ar[dr] \ar[ur] & \\ F \ar[ur] \ar@/_2pc/[rr]_{\gamma} && G \ar[ur] \ar@/_2pc/[rr]_{\delta} && H \\ }$

1) Show that there is a rudimentary Mayer-Vietoris exact sequence

$\displaystyle D \to B\oplus G \to E.$ $$\displaystyle D \to B\oplus G \to E.$$

2) Show that there is defined an isomorphism

$\displaystyle \mathrm{Ker}(\beta)/\mathrm{Im}(\alpha) \cong \mathrm{Ker}(\delta)/\mathrm{Im}(\gamma).$ $$\displaystyle \mathrm{Ker}(\beta)/\mathrm{Im}(\alpha) \cong \mathrm{Ker}(\delta)/\mathrm{Im}(\gamma).$$