
theorem
  for T being RelStr, A being upper Subset of T, B being directed Subset of T
  holds A /\ B is directed
proof
  let T be RelStr, A be upper Subset of T, B be directed Subset of T;
  let x,y be Element of T such that
A1: x in A /\ B and
A2: y in A /\ B;
A3: x in B by A1,XBOOLE_0:def 4;
  y in B by A2,XBOOLE_0:def 4;
  then consider z being Element of T such that
A4: z in B and
A5: x <= z and
A6: y <= z by A3,WAYBEL_0:def 1;
  take z;
  x in A by A1,XBOOLE_0:def 4;
  then z in A by A5,WAYBEL_0:def 20;
  hence z in A /\ B by A4,XBOOLE_0:def 4;
  thus thesis by A5,A6;
end;
