
theorem
  for R being locally_directed non empty Poset, x,y being Element of R
  st (ex z being Element of R st z <= x & z <= y) holds ex u being Element of R
  st x <= u & y <= u
proof
  let R be locally_directed non empty Poset, x,y be Element of R;
  assume ex z being Element of R st z <= x & z <= y;
  then consider z being Element of R such that
A1: z <= x and
A2: z <= y;
  reconsider x1 = x,y1 = y as Element of R;
  CComp(z) = CComp(y) by A2,Th4;
  then
A3: y in CComp(z) by EQREL_1:20;
  CComp z = CComp(x) by A1,Th4;
  then x in CComp(z) by EQREL_1:20;
  then ex u being Element of R st u in CComp(z) & x1 <= u & y1 <= u by A3,
WAYBEL_0:def 1;
  hence thesis;
end;
