Just to shed some light,
The 4500 mA battery is the total capacity rating of the battery. (the size of a car's gas tank)
The UA5 is labled for how much current it draws. (i.e. mpg in car)
So the UA5 uses 0.45 amps per hour. and the 4500mA battery can supply 4.5amps per hour (total).
mAH = milliamp hours.
So theoreticaly, the UA5 should last approx 10 hrs on a full charge.
The good thing about Lithium Ions, they hold the voltage steady over the life of the battery. THey do lose capacity over life time.
NiMH lose voltage level over usage. If the voltage drops to low, the device will shut off.
When a LiO runs out of charge (fuel), it simply shuts off.
Hope this helps others.