reserve A, B, X, Y for set;

theorem
  for N being antisymmetric with_infima RelStr, D, E being Subset of N
  for X being upper Subset of N st D misses X holds D "/\" E misses X
proof
  let N be antisymmetric with_infima RelStr, D, E be Subset of N, X be upper
  Subset of N such that
A1: D /\ X = {};
  assume (D "/\" E) /\ X <> {};
  then consider k being object such that
A2: k in (D "/\" E) /\ X by XBOOLE_0:def 1;
  reconsider k as Element of N by A2;
A3: D "/\" E = { d "/\" e where d, e is Element of N : d in D & e in E } & k
  in D "/\" E by A2,XBOOLE_0:def 4,YELLOW_4:def 4;
A4: k in X by A2,XBOOLE_0:def 4;
  consider d, e being Element of N such that
A5: k = d "/\" e and
A6: d in D and
  e in E by A3;
  d "/\" e <= d by YELLOW_0:23;
  then d in X by A4,A5,WAYBEL_0:def 20;
  hence thesis by A1,A6,XBOOLE_0:def 4;
end;
