
theorem
  for L being antisymmetric with_infima RelStr for A being Subset of L,
  B being non empty Subset of L holds A is_coarser_than A "/\" B
proof
  let L be antisymmetric with_infima RelStr, A be Subset of L, B be non empty
  Subset of L;
  consider b being object such that
A1: b in B by XBOOLE_0:def 1;
  reconsider b as Element of B by A1;
  let q be Element of L such that
A2: q in A;
  take q "/\" b;
  thus q "/\" b in A "/\" B by A2;
  thus thesis by YELLOW_0:23;
end;
