Why is there horse in a Military Funeral?

If you have ever seen a soldier’s funeral, the military way, which is very honorable. There will always be a riderless horse but with a saddle on it’s back. If nobody is going to ride it then why put a saddle on it? And why is there a horse in a ceremony anyway? It came from an ancient Roman practice, a soldier and his horse were trained to fight as one, they would go through so many battles and be together for years. If the soldier would die, the Romans thought the horse should follow it’s master.  The horse would be there in the funeral but not as showing, but to be actually buried with the soldier. The two would go into another journey in the afterlife. Today, the practice isn’t still in use, the horse is still there as a symbolic representation of honor, courage, they don’t actually kill or bury the horse alive.