
theorem
  for L being with_suprema Poset st for X being directed Subset of L, x
  being Element of L holds x "/\" sup X = sup ({x} "/\" X) holds for X being
Subset of L, x being Element of L st ex_sup_of X,L holds x "/\" sup X = sup ({x
  } "/\" finsups X)
proof
  let L be with_suprema Poset such that
A1: for X being directed Subset of L, x being Element of L holds x "/\"
  sup X = sup ({x} "/\" X);
  let X be Subset of L, x be Element of L;
  assume ex_sup_of X,L;
  hence x "/\" sup X = x "/\" sup finsups X by WAYBEL_0:55
    .= sup ({x} "/\" finsups X) by A1;
end;
