Proof:

In the diagrams below, the Left hand square depicts a push-out square of inclusions which goes over to a push-out square of complexes on the right:

$\begin{CD} U\cap V @> i_1 >> U \\ @V{i_2}VV @VV{j_1}V\\ V @> j_2 >> U\cup V \\ \end{CD}$ $\begin{CD} S(U\cap V) @> i_1 >> S(U) \\ @V{i_2}VV @VV{j_1}V\\ S(V) @> j_2 >> S^{\;{\cal U}}(U\cup V) \\ \end{CD}$

The reader may check that the latter may be recast as a short exact sequence of chain complexes namely

$$\begin{CD} 0 @> >> S(U\cap V)@> (i_1, -i_2) >> S(U)\oplus S(V) @> j_1 + j_2 >> S^{\;{\cal U}}(U\cap V)@> >> 0. \end{CD} \eqno(34.11)$$

The corresponding long exact sequence in homology gives

$$\begin{CD} @> >> H_n(U\cap V) @> (\kappa^{\prime}, -\kappa^{\pri... ... @> Q_n >> H^{\;{\cal U}}_n(U\cup V) @>{D_n} >> H_{n-1}(U\cap V) @> >> \end{CD}$$

The definition of $\kappa^{\prime}, \kappa^{\prime\prime}$ and exercise 6 enables us to replace $Q_n$ and $D_n$ by the composites

$$\begin{CD} q_n \;: \; H_n(U)\oplus H_n(V) @> Q_n >> H^{\;{\cal U}}_n(U\cup V) @> \lambda >> H_n(U\cup V) \end{CD} \phantom{XXXXXXXXXXXX}$$

$$\phantom{X}$$

$$\begin{CD} \phantom{XXXXXX.XX}\delta_n \;: \; H_n(U\cap V) @> \la... ...cal U}}_n(U\cup V) @> D_n >> H_n(U\cap V)\phantom{XXXXXXXXXX..}(34.12) \end{CD}$$

where $\lambda$ is the isomorphism given by theorem (34.6). The final result is the Mayer Vietoris sequence stated in the theorem. The second part is clear from (29.18). $\square$