
theorem Th4:
  for L being antisymmetric reflexive with_suprema RelStr, x being
  Element of L holds uparrow x = {x} "\/" [#]L
proof
  let L be antisymmetric reflexive with_suprema RelStr, x be Element of L;
A1: {x} "\/" [#]L = {x "\/" s where s is Element of L : s in [#] L} by
YELLOW_4:15;
  thus uparrow x c= {x} "\/" [#]L
  proof
    let q be object;
    assume
A2: q in uparrow x;
    then reconsider q1 = q as Element of L;
    x <= q1 by A2,WAYBEL_0:18;
    then x "\/" q1 = q1 by YELLOW_0:24;
    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 <= x "\/" z by YELLOW_0:22;
  hence thesis by A3,WAYBEL_0:18;
end;
