
theorem
  for L being non empty lower-bounded antisymmetric RelStr, x being
  Element of L, R being auxiliary(iv) Relation of L holds {Bottom L, x} is
  strict_chain of R
proof
  let L be non empty lower-bounded antisymmetric RelStr, x be Element of L, R
  be auxiliary(iv) Relation of L;
  let a, y be set;
  assume that
A1: a in {Bottom L, x} and
A2: y in {Bottom L, x};
A3: y = Bottom L or y = x by A2,TARSKI:def 2;
  a = Bottom L or a = x by A1,TARSKI:def 2;
  hence thesis by A3,WAYBEL_4:def 6;
end;
