Later, helmets are randomly checked again. I have met some of those involved in this testing and watched them at work. The ASTM F1163 equestrian standard allows the least amount of stretch to the retention system - that's your jaw strap or harness. Each standard tests the helmets by dropping them from a specific height, at a specific speed, onto a steel plate. Some drop test onto a flat surface, some onto a hemisphere or curb-like surface.
Equestrian helmets are also drop tested onto a steel anvil with a ninety degree edge.
It's that simple. If you do, you are risking years of enjoyment with horses for that one ride. If they don't, find a new instructor. There are two groups which certify instructors in the U.S. - both support helmets. A shell hunt cap or an old Pony Club standard hat is NOT a protective hat.