A problem of guaranteed result optimization is considered for a control system described by an ordinary differential equation and for a performance functional that depends continuously on the trajectory of the system. The values of the control and of the disturbance satisfy compact geometric constraints. It is also assumed that the realization of the disturbance is subject to an unknown functional constraint from a given set of constraints that are compact in a Lebesgue space. We give a proof of the earlier announced result that in this problem the optimal guaranteed result in the class of full-memory strategies coincides with the value of the optimal guaranteed result in the class of quasi-strategies.