
theorem
  for L being Semilattice, D1, D2 being Subset of L holds uparrow ((
  uparrow D1) "/\" (uparrow D2)) = uparrow (D1 "/\" D2)
proof
  let L be Semilattice, D1, D2 be Subset of L;
A1: uparrow (D1 "/\" D2) = {s where s is Element of L: ex z being Element of
  L st s >= z & z in D1 "/\" D2} by WAYBEL_0:15;
  thus uparrow ((uparrow D1) "/\" (uparrow D2)) c= uparrow (D1 "/\" D2) by Th63
;
  let q be object;
  assume q in uparrow (D1 "/\" D2);
  then consider s being Element of L such that
A2: q = s and
A3: ex z being Element of L st s >= z & z in D1 "/\" D2 by A1;
A4: uparrow ((uparrow D1) "/\" (uparrow D2)) = {x where x is Element of L:
  ex t being Element of L st x >= t & t in (uparrow D1) "/\" (uparrow D2)} by
WAYBEL_0:15;
A5: D1 is Subset of uparrow D1 & D2 is Subset of uparrow D2 by WAYBEL_0:16;
  consider x being Element of L such that
A6: s >= x and
A7: x in D1 "/\" D2 by A3;
  ex a, b being Element of L st x = a "/\" b & a in D1 & b in D2 by A7;
  then x in (uparrow D1) "/\" (uparrow D2) by A5;
  hence thesis by A4,A2,A6;
end;
