
theorem Th61:
  for L being transitive antisymmetric with_infima RelStr for D1,
  D2 being Subset of L holds downarrow ((downarrow D1) "/\" (downarrow D2)) c=
  downarrow (D1 "/\" D2)
proof
  let L be transitive antisymmetric with_infima RelStr, D1, D2 be Subset of L;
A1: 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;
  let y be object;
  assume y in downarrow ((downarrow D1) "/\" (downarrow D2));
  then consider x being Element of L such that
A2: y = x and
A3: ex t being Element of L st x <= t & t in (downarrow D1) "/\" (
  downarrow D2) by A1;
  consider x1 being Element of L such that
A4: x <= x1 and
A5: x1 in (downarrow D1) "/\" (downarrow D2) by A3;
  consider a, b being Element of L such that
A6: x1 = a "/\" b and
A7: a in downarrow D1 and
A8: b in downarrow D2 by A5;
  downarrow D2 = {s2 where s2 is Element of L: ex z being Element of L st
  s2 <= z & z in D2} by WAYBEL_0:14;
  then consider B1 being Element of L such that
A9: b = B1 and
A10: ex z being Element of L st B1 <= z & z in D2 by A8;
  consider b1 being Element of L such that
A11: B1 <= b1 and
A12: b1 in D2 by A10;
  downarrow D1 = {s1 where s1 is Element of L: ex z being Element of L st
  s1 <= z & z in D1} by WAYBEL_0:14;
  then consider A1 being Element of L such that
A13: a = A1 and
A14: ex z being Element of L st A1 <= z & z in D1 by A7;
  consider a1 being Element of L such that
A15: A1 <= a1 and
A16: a1 in D1 by A14;
  x1 <= a1 "/\" b1 by A6,A13,A9,A15,A11,YELLOW_3:2;
  then
A17: downarrow (D1 "/\" D2) = {s where s is Element of L: ex z being Element
  of L st s <= z & z in D1 "/\" D2} & x <= a1 "/\" b1 by A4,ORDERS_2:3
,WAYBEL_0:14;
  a1 "/\" b1 in D1 "/\" D2 by A16,A12;
  hence thesis by A2,A17;
end;
