
theorem
  for L being antisymmetric with_suprema RelStr for A being Subset of L,
  B being non empty Subset of L holds A is_finer_than A "\/" B
proof
  let L be antisymmetric with_suprema RelStr, A be Subset of L, B be non empty
  Subset of L;
  let q be Element of L such that
A1: q in A;
  set b = the Element of B;
  take q "\/" b;
  thus q "\/" b in A "\/" B by A1;
  thus thesis by YELLOW_0:22;
end;
