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;
