reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;
reserve SA for non empty semiautomaton over F;

theorem Th15:
  for Q being Subset of SA holds w in left-Lang(Q) iff Q meets w
  -succ_of (the InitS of SA, SA)
proof
  let Q be Subset of SA;
  thus w in left-Lang(Q) implies Q meets w-succ_of (the InitS of SA, SA)
  proof
    assume w in left-Lang(Q);
    then ex w9 st w9 = w & Q meets w9-succ_of (the InitS of SA, SA);
    hence thesis;
  end;
  thus thesis;
end;
