
theorem
  for L being sup-Semilattice, I being lower directed Subset of L holds
  I "\/" I = I
proof
  let L be sup-Semilattice, I be lower directed Subset of L;
  thus I "\/" I c= I
  proof
    let x be object;
    assume x in I "\/" I;
    then consider y, z being Element of L such that
A1: x = y "\/" z and
A2: y in I & z in I;
    consider t being Element of L such that
A3: t in I and
A4: t >= y & t >= z by A2,WAYBEL_0:def 1;
    y "\/" z <= t by A4,YELLOW_0:22;
    hence thesis by A1,A3,WAYBEL_0:def 19;
  end;
  thus thesis by YELLOW_4:13;
end;
