
theorem
  for L being up-complete LATTICE st for X being Subset of L, x being
  Element of L holds x "/\" sup X = sup ({x} "/\" finsups X) holds for X being
non empty directed Subset of L, x being Element of L holds x "/\" sup X = sup (
  {x} "/\" X)
proof
  let L be up-complete LATTICE such that
A1: for X being Subset of L, x being Element of L holds x "/\" sup X =
  sup ({x} "/\" finsups X);
  let X be non empty directed Subset of L, x be Element of L;
  reconsider T = {x} as non empty directed Subset of L by WAYBEL_0:5;
A2: ex_sup_of T "/\" X,L by WAYBEL_0:75;
A3: {x} "/\" finsups X = {x "/\" y where y is Element of L : y in finsups X}
  by YELLOW_4:42;
A4: {x} "/\" finsups X is_<=_than sup ({x} "/\" X)
  proof
    let q be Element of L;
A5: x <= x;
    assume q in {x} "/\" finsups X;
    then consider y being Element of L such that
A6: q = x "/\" y and
A7: y in finsups X by A3;
    consider Y being finite Subset of X such that
A8: y = "\/"(Y,L) and
A9: ex_sup_of Y,L by A7;
    consider z being Element of L such that
A10: z in X and
A11: z is_>=_than Y by WAYBEL_0:1;
    "\/"(Y,L) <= z by A9,A11,YELLOW_0:30;
    then
A12: x "/\" y <= x "/\" z by A8,A5,YELLOW_3:2;
    x in {x} by TARSKI:def 1;
    then x "/\" z <= sup ({x} "/\" X) by A2,A10,YELLOW_4:1,37;
    hence q <= sup ({x} "/\" X) by A6,A12,YELLOW_0:def 2;
  end;
  ex_sup_of T "/\" finsups X,L by WAYBEL_0:75;
  then sup ({x} "/\" finsups X) <= sup ({x} "/\" X) by A4,YELLOW_0:30;
  then
A13: x "/\" sup X <= sup ({x} "/\" X) by A1;
  ex_sup_of X,L by WAYBEL_0:75;
  then sup ({x} "/\" X) <= x "/\" sup X by A2,YELLOW_4:53;
  hence thesis by A13,ORDERS_2:2;
end;
