War Eagle, or anyone -- Help!

I among the many confused folks about what ammo is safe to shoot in my early, apparently not heat treated cylinder, Model 1902, 1 st change 32/20 hand ejector revolver of about 1903-1905.

I have a full box (green box) of factory Remington Hi-Speed cartridges marked on the box: 32-20 WIN., 100 GRAIN LEAD, INDEX 4232 -- are these OK to shoot in my revolver?

I believe this is the ammo used by the previous owner, but still wonder. I don't want to damage this fine old gun, or myself.

Thanks for your help,

IMC