is made 1 that is when a predetermined time has elapsed after a state of engine stop and start of the engine leaning of the air fuel ratio is started by the present calculation unit the instability index ind_sta is calculated which is specifically shown in fig2 however because it is same with that of example 1 detail description thereof will be omitted by the present calculation unit the basic fuel injection quantity correction value f_hos is calculated which is specifically shown in fig2 when fp_cont 1 0 f_hos is made 1 0 when fp_cont 1 1 f_hos is reduced by k 1 _f stepwise until ind_sta u2267 k 1 _sta is achieved when fp_cont 1 1 and fp_cont 2 1 f_hos maintains a previous value k 1 _sta is made an instability index value equivalent to a target air fuel ratio also it may be changed based on the operation condition of the engine it may be changed also based on a change deterioration of the light off performance of the catalyst more specifically k 1 _sta is increased according to deterioration of the light off performance of the catalyst k 1 _f is a value that determines a leaning speed and is determined taking responsiveness of the engine and the like also into account by the present calculation unit the control permit flag 2 fp_cont 2 is calculated which is specifically shown in fig2 when fp_cont 1 0 and ind_sta u2267 k 1 _sta.