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;
reserve TS for non empty transition-system over Lex(E) \/ {<%>E};
reserve SA for non empty semiautomaton over Lex(E) \/ {<%>E};

theorem Th32:
  w-succ_of ({ <%>E-succ_of (X, SA) }, _bool SA) = { w-succ_of (X, SA) }
proof
  set TS = the transition-system of SA;
  set Es = <%>E-succ_of (X, SA);
  the transition-system of _bool SA = _bool TS by Def3;
  hence w-succ_of ({ Es }, _bool SA) = w-succ_of ({ Es }, _bool TS) by
REWRITE3:105
    .= w-succ_of ({ <%>E-succ_of (X, TS) }, _bool TS) by REWRITE3:105
    .= { {}^w-succ_of (X, TS) } by Th31
    .= { w-succ_of (X, TS) }
    .= { w-succ_of (X, SA) } by REWRITE3:105;
end;
