objects and advantages of the invention are achieved by: a first process of the present invention which comprises, contacting fibers with hydrogen peroxide under conditions which provide oxidative bleaching of the fibers to produce bleached fibers in contact with unspent hydrogen peroxide; adding to the bleached fibers in contact with unspent hydrogen peroxide (from the previous step), a material which combines with hydrogen peroxide to form a reductive bleaching agent (e.g. thiourea) in an amount sufficient to produce a reductive bleaching media; and maintaining the oxidatively bleached fibers in said reductive bleaching media under conditions providing reductive bleaching of the bleached fibers, and; a second process of the present invention which comprises adding to the bleached fibers in contact with unspent hydrogen peroxide (from the previous step), an inactivating material in an amount at least sufficient to inactivate all of the unspent hydrogen peroxide to form an inactivated media; and subsequent to the inactivation of all the unspent hydrogen peroxide, reductively bleaching the bleached fibers by addition of a reductive bleaching agent to the inactivated media