
theorem Th12:
  for P being with_suprema Poset, x, y being Element of P holds (
  wayabove x)"\/"(wayabove y) c= uparrow (x"\/"y)
proof
  let R be with_suprema Poset, x, y be Element of R;
  {x}"\/"{y} = {x"\/"y} & (uparrow x)"\/"(uparrow y) c= uparrow ((uparrow
  x) "\/"(uparrow y)) by WAYBEL_0:16,YELLOW_4:19;
  then
A1: (uparrow x)"\/"(uparrow y) c= uparrow (x"\/"y) by YELLOW_4:35;
  wayabove x c= uparrow x & wayabove y c= uparrow y by WAYBEL_3:11;
  then (wayabove x)"\/"(wayabove y) c= (uparrow x)"\/"(uparrow y) by
YELLOW_4:21;
  hence thesis by A1;
end;
