
theorem
  for A being non empty RelStr,
    a1, a2 being Element of A st A is strongly_connected holds
      a1 <~ a2 or a1 =~ a2 or a1 >~ a2
proof
  let A be non empty RelStr;
  let a1, a2 be Element of A;
  assume A1: A is strongly_connected;
  [a1,a2] in the InternalRel of A or
    [a2,a1] in the InternalRel of A by A1, RELAT_2:def 7;
  then A2: a1 <= a2 or a1 >= a2 by ORDERS_2:def 5;
  assume not (a1 <~ a2 or a1 =~ a2 or a1 >~ a2);
  hence contradiction by A2;
end;
