Functional operators, simpler than in /1-3/, are determined in connection with games of encounter at a prescribed instant. These operators or their analogs can be used to obtain many of the fundamental results in /1-3/, in particular, iteration methods can be constructed, converging to the game's value function. As one more illustration of the capabilities of the method developed in /1-3/, the operators determined below are used to give a new proof of the well-known result /4/ on the identification of the game's value function by means of the so-called main equation.