Exact Sequences and short exact sequences - basic question

In Dummit and Foote Section 10.5 Exact Sequences (see attachment) we read the following on page 379:

"Note that any exact sequence can be written as a succession of short exact sequences since to say

$\displaystyle X \longrightarrow Y \longrightarrow Z $

[where the homomorphisms involved are as follows; $\displaystyle \alpha \ : \ X \longrightarrow Y $ and $\displaystyle \beta \ : \ Y \longrightarrow Z $ ]

is exact at Y is the same as saying that the sequence

$\displaystyle 0 \longrightarrow \alpha (X) \longrightarrow Y/ {ker \beta} \longrightarrow 0 $

is a short exact sequence.

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I am trying to get an understanding of this statement.

Can someone please demonstrate formally that this is true.

To enable me to get an understanding of this it would help enormously if someone could devise an example of this.

Peter

Hi Peter,
I think you would have understood immediately if the authors had indicated the homomorphisms in the short exact sequence. Here they are:

Attachment 29022