
theorem Th2:
  for L be RelStr for A,B be set holds A,B
  form_upper_lower_partition_of L implies A misses B
proof
  let L be RelStr;
  let A,B be set;
  assume that
A1: A,B form_upper_lower_partition_of L and
A2: A meets B;
  consider x be object such that
A3: x in A /\ B by A2,XBOOLE_0:4;
A4: x in B by A3,XBOOLE_0:def 4;
A5: x in A by A3,XBOOLE_0:def 4;
  A \/ B = the carrier of L by A1;
  then reconsider x as Element of L by A5,XBOOLE_0:def 3;
  x < x by A1,A5,A4;
  hence contradiction;
end;
