
theorem Th8:
  for R being discrete non empty Poset holds R is locally_directed
proof
  let R be discrete non empty Poset;
  let C be Component of R;
  consider x being Element of R such that
A1: C = {x} by Th7;
  for x,y being Element of R st x in C & y in C ex z being Element of R st
  z in C & x <= z & y <= z
  proof
    let x1,y1 be Element of R such that
A2: x1 in C and
A3: y1 in C;
    take x1;
    x1 = x by A1,A2,TARSKI:def 1;
    hence thesis by A1,A3,TARSKI:def 1;
  end;
  hence thesis by WAYBEL_0:def 1;
end;
