Approaching-deviation game task at $m$ aim sets for step by step changing systems

An approaching-deviation game task of m aim sets and step by step changing system is considered in the class of quazistrategies. In this case Lepshits’s condition is replaced by the condition of existing general singularity and the saddle point existing condition in thee little game is replaced by a weaker condition of resource collection coordination. An affirmation is proved which is analogous to the theorem of the existing game prize in differential games.