( L 2 N ,S S C , V ) := { ( P, v ) | P ∈ L 2 N ,S S C and v is a marked point on P such that P v L is a Schr¨oder path and the succeeding step of v is not U } . By definition, v must be an even integer point on the x -axis, and either it is the right end of P or its succeeding step is D or L2 . We claim that the generating function of ( L 2 N ,S S C , V ) is exactly CF SC ,x ( x, u, w, y ), where the exponent of y counts the number of flaw steps D and prairies L 2 on the subpath P v R (or the semi-length of all weakly flaw steps on P v R )—we will call these steps the y -related steps for short. For independent on y , CF a should be the generating function of those ( P, v ) ∈ ( L 2 N ,S S C , V ) that have v locating at the end of P . It is correct. Because P v L is a normal Schr¨oder path and the fact P = P v L