
theorem
  for L being with_infima antisymmetric RelStr for X being Subset of L,
  Y being non empty Subset of L holds X c= uparrow (X "/\" Y)
proof
  let L be with_infima antisymmetric RelStr, X be Subset of L, Y be non empty
  Subset of L;
  consider y being object such that
A1: y in Y by XBOOLE_0:def 1;
  reconsider y as Element of Y by A1;
  let q be object;
  assume
A2: q in X;
  then reconsider X1 = X as non empty Subset of L;
  reconsider x = q as Element of X1 by A2;
  ex s being Element of L st x >= s & y >= s & for c being Element of L st
  x >= c & y >= c holds s >= c by LATTICE3:def 11;
  then
A3: x "/\" y <= x by LATTICE3:def 14;
  uparrow (X "/\" Y) = {s where s is Element of L: ex y being Element of L
  st s >= y & y in X "/\" Y} & x "/\" y in X1 "/\" Y by WAYBEL_0:15;
  hence thesis by A3;
end;
