I'm facing problem in proving this theorem that shows the existence of free products with amalgamated subgroups.

$\displaystyle A\;free \;product \;with \;amalgamated \;subgroup \;exists.$

Although the book gave the proof, but it is very very short and tedious.
I found some similar proof online, but it's still can't clarify everything.

I need a detail proof.

Thank you for those who are patient to explain to me.