
theorem Th3:
  for L being non empty lower-bounded antisymmetric RelStr for x
being Element of L holds (L is with_infima implies Bottom L "/\" x = Bottom L)
  & (L is with_suprema reflexive transitive implies Bottom L "\/" x = x)
proof
  let L be non empty lower-bounded antisymmetric RelStr;
  let x be Element of L;
  thus L is with_infima implies Bottom L "/\" x = Bottom L
  proof
    assume L is with_infima;
    then Bottom L <= Bottom L "/\" x & Bottom L "/\" x <= Bottom L by
YELLOW_0:23,44;
    hence thesis by ORDERS_2:2;
  end;
  assume
A1: L is with_suprema;
  then
A2: x <= Bottom L "\/" x by YELLOW_0:22;
  assume L is reflexive transitive;
  then
A3: x <= x;
  Bottom L <= x by YELLOW_0:44;
  then Bottom L "\/" x <= x by A1,A3,YELLOW_0:22;
  hence thesis by A2,ORDERS_2:2;
end;
