When we are feeling our worst about losing anyone that we love, we have to cope somehow and keep going and do the right thing. The only way I've ever been able to do this is to fully believe that when my pet is in pain and not able to live the way they always have that they will be at peace when they pass away. I have to envision an end to their pain and suffering. If you approach it as the ultimate act of love, it is doable.
I don't know if this really applies to people, though. I don't think I believe in euthanizing people. With people, there is often a hope in a medical miracle. Our bodies can fight. With a pet, they are so small and veterinary medicine is nowhere near as advanced, that when they start to slide it's often impossible to bring them back.
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"I never made "Who's Who"- but sure as hell I made "What's That??..."