
theorem
  for L being Semilattice, D1, D2 being Subset of L holds downarrow ((
  downarrow D1) "/\" (downarrow D2)) = downarrow (D1 "/\" D2)
proof
  let L be Semilattice, D1, D2 be Subset of L;
A1: downarrow (D1 "/\" D2) = {s where s is Element of L: ex z being Element
  of L st s <= z & z in D1 "/\" D2} by WAYBEL_0:14;
  thus downarrow ((downarrow D1) "/\" (downarrow D2)) c= downarrow (D1 "/\" D2
  ) by Th61;
  let q be object;
  assume q in downarrow (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: downarrow ((downarrow D1) "/\" (downarrow D2)) = {x where x is Element
  of L: ex t being Element of L st x <= t & t in (downarrow D1) "/\" (downarrow
  D2)} by WAYBEL_0:14;
A5: D1 is Subset of downarrow D1 & D2 is Subset of downarrow 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 (downarrow D1) "/\" (downarrow D2) by A5;
  hence thesis by A4,A2,A6;
end;
