I came across the claim that there is an exact sequence $K_1\Lambda\to K_1\Lambda_2\oplus K_1\Lambda_2\to K_1\Lambda'\to K_0\Lambda\to K_0\Lambda_1\oplus K_0\Lambda_2\to K_0\Lambda'$ while reading Milnor's Introduction to Algebraic K-theory's page 28.
The homomorphisms in $K_\alpha\Lambda\to K_\alpha\Lambda_1\oplus K_\alpha\Lambda_2\to K_\alpha\Lambda'$ are defined by $x\to(i_{1*}x,i_{2*}x)$ and $(y,z)\to j_{1*}y-j_{2*}z$. The connecting homomorphism $\partial:K_1\Lambda'\to K_0\Lambda$ is defined by taking a representation of an element $x\in K_1\Lambda$ and seeing it as an isomorphism $h$ from $j_{1\#}\Lambda_1^n$ to $j_{2\#}\Lambda_2^n$, thereby constructing the projective module $M=M(\Lambda_1^n,\Lambda_2^n,h)$ over $\Lambda$ and let $\partial(x)=[M]-[\Lambda^n]\in K_0\Lambda$.
Milnor omits the proof of the exactness of the sequene so I tried to prove it myself. I could verify all the other steps except that $Ker(\partial)$ is a subset of the image of the map from $K_1\Lambda_1\oplus K_1\Lambda_2$ and that $Im(\partial)$ contains the kernel of the map from $K_0\Lambda$. Any ideas how to do this?