
theorem Th11:
  for L being non empty reflexive antisymmetric RelStr
  for x being Element of L holds
  waybelow x c= downarrow x & wayabove x c= uparrow x
proof
  let L be non empty reflexive antisymmetric RelStr, x be Element of L;
  hereby
    let a be object;
    assume a in waybelow x;
    then consider y being Element of L such that
A1: a = y and
A2: y << x;
    y <= x by A2,Th1;
    hence a in downarrow x by A1,WAYBEL_0:17;
  end;
  let a be object;
  assume a in wayabove x;
  then consider y being Element of L such that
A3: a = y and
A4: y >> x;
  x <= y by A4,Th1;
  hence thesis by A3,WAYBEL_0:18;
end;
