theorem
  for TS being transition-system over F st not <%>E in rng dom (the Tran
  of TS) holds not x, <%>E -->. y, TS
proof
  let TS be transition-system over F such that
A1: not <%>E in rng dom (the Tran of TS);
  assume x, <%>E -->. y, TS;
  then [[x, <%>E], y] in the Tran of TS;
  then [x, <%>E] in dom (the Tran of TS) by XTUPLE_0:def 12;
  hence contradiction by A1,XTUPLE_0:def 13;
end;
