
theorem
  for L being antisymmetric with_suprema RelStr, A, B being Subset of L
  holds A "\/" B is_coarser_than A
proof
  let L be antisymmetric with_suprema RelStr, A, B be Subset of L;
  let q be Element of L;
  assume q in A "\/" B;
  then consider x, y being Element of L such that
A1: q = x "\/" y and
A2: x in A and
  y in B;
  take x;
  thus x in A by A2;
  thus thesis by A1,YELLOW_0:22;
end;
