reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th40:
  for L being with_infima Poset for I, J being Ideal of L holds I
  /\ J is Ideal of L
proof
  let L be with_infima Poset;
  let I, J be Ideal of L;
  set i = the Element of I,j = the Element of J;
  set a = i"/\"j;
  a <= j by YELLOW_0:23;
  then
A1: a in J by WAYBEL_0:def 19;
  a <= i by YELLOW_0:23;
  then a in I by WAYBEL_0:def 19;
  hence thesis by A1,WAYBEL_0:27,44,XBOOLE_0:def 4;
end;
