The problem of cooperation in repeated and multistage games is considered. The strong equilibrium (equilibrium stable against deviations of coalitions) with payoffs which can be attained under cooperation is constructed for a wide class of such games. The new solution concept based on solutions of stage games is introduced and in some cases this solution is a subset of the core defined for repeated and multistage games in a classical way. It is also proved that this newly introduced solution concept is strongly time consistent. The strong time consistency of the solution is a very important property since in case it does not take place players in reality in some time instant in subgame on cooperative trajectory may switch from the previously selected optimal solution to any other optimal solution in the subgame and as result realize the solution which will not be optimal in the whole game.