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};
reserve A for non empty automaton over Lex(E) \/ {<%>E};
reserve P for Subset of A;

theorem Th36:
  w-succ_of ({ <%>E-succ_of (X, A) }, _bool A) = { w-succ_of (X, A ) }
proof
  set SA = the semiautomaton of A;
  set Es = <%>E-succ_of (X, A);
  the semiautomaton of _bool A = _bool SA by Def6;
  hence w-succ_of ({ Es }, _bool A) = w-succ_of ({ Es }, _bool SA) by
REWRITE3:105
    .= w-succ_of ({ <%>E-succ_of (X, SA) }, _bool SA) by REWRITE3:105
    .= { w-succ_of (X, SA) } by Th32
    .= { w-succ_of (X, A) } by REWRITE3:105;
end;
