Two theorems about Galiukschov semicontextual languages

We assume the reader familiar with the basic notions of formal language theory (from [3], for example) and we specify only some notions about the semicontextual grammars introduced in [1], under linguistic motivations. A semicontextual grammar is a triple G = (V, B, P), where V is a nonempty finite alphabet, B is a finite language over V and P is a finite set of rewriting rules of the form xy -> xzy, x, y, z being non-null strings over V. If w -= uxyv and w' = = uxzyv are two strings in V* (V* is the free monoid generated by V under the concatenation operation and the null element X) and xy -* xzy is a rule in P, then we write vv => w'. We denote by =>* the reffexive transitive closure of the relation => and define the language generated by G as