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;
reserve A for non empty automaton over F;
reserve p, q for Element of A;

theorem
  Lang(A) = left-Lang(the FinalS of A)
proof
A1: w in Lang(A) implies w in left-Lang(the FinalS of A)
  proof
    assume w in Lang(A);
    then w-succ_of (the InitS of A, A) meets (the FinalS of A) by Th19;
    hence thesis;
  end;
  w in left-Lang(the FinalS of A) implies w in Lang(A)
  proof
    assume w in left-Lang(the FinalS of A);
    then the FinalS of A meets w-succ_of (the InitS of A, A) by Th15;
    hence thesis by Th19;
  end;
  hence thesis by A1,SUBSET_1:3;
end;
