
theorem Th4:
  for L being non empty upper-bounded antisymmetric RelStr for x
  being Element of L holds (L is with_infima transitive reflexive implies Top L
  "/\" x = x) & (L is with_suprema implies Top L "\/" x = Top L)
proof
  let L be non empty upper-bounded antisymmetric RelStr, x be Element of L;
  thus L is with_infima transitive reflexive implies Top L "/\" x = x
  proof
    assume
A1: L is with_infima transitive reflexive;
    then x "/\" x <= Top L "/\" x by Th1,YELLOW_0:45;
    then
A2: x <= Top L "/\" x by A1,YELLOW_0:25;
    Top L "/\" x <= x by A1,YELLOW_0:23;
    hence thesis by A2,ORDERS_2:2;
  end;
  assume L is with_suprema;
  then Top L "\/" x <= Top L & Top L <= Top L "\/" x by YELLOW_0:22,45;
  hence thesis by ORDERS_2:2;
end;
