For each admissible map l?, the value v (r-I (B)) stands as a "measure9' of the set r-' (B) of all solutions X E to the inclusion r (X) E B.

Let 0 = v (0). An admissible map l? is said to be v-essential (in M ) if

rlA = I'llA and v (F-' (B)) # v (r'-' (B)) . Also consider an equivalence relation on M with

We are interested in the case when the equivalence classes contain only v-essential maps or only v-inessential maps. A sufficient condition to have such a case is the following one:

(H) if I' m F', then there is an homotopy q : E X [0, l] -+ O and a function v : E + [O, l] such that

and 1 for X E CV

v (X) = 0 for X E A,

F' and v (IT' (B)) # v (F'-' (B)) . ( 9 4

Proof. The necessity follows from the definition of the v-inessential maps and condition (A).