
theorem Th5:
  for L being antisymmetric reflexive with_infima RelStr, x being
  Element of L holds downarrow x = {x} "/\" [#]L
proof
  let L be antisymmetric reflexive with_infima RelStr, x be Element of L;
A1: {x} "/\" [#]L = {x "/\" s where s is Element of L : s in [#] L} by
YELLOW_4:42;
  thus downarrow x c= {x} "/\" [#]L
  proof
    let q be object;
    assume
A2: q in downarrow x;
    then reconsider q1 = q as Element of L;
    x >= q1 by A2,WAYBEL_0:17;
    then x "/\" q1 = q1 by YELLOW_0:25;
    hence thesis by A1;
  end;
  let q be object;
  assume q in {x} "/\" [#]L;
  then consider z being Element of L such that
A3: q = x "/\" z and
  z in [#]L by A1;
  x "/\" z <= x by YELLOW_0:23;
  hence thesis by A3,WAYBEL_0:17;
end;
