
theorem Th29:
  for L being with_suprema antisymmetric RelStr for X being Subset
  of L, Y being non empty Subset of L holds X c= downarrow (X "\/" Y)
proof
  let L be with_suprema 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 10;
  then
A3: x <= x "\/" y by LATTICE3:def 13;
  downarrow (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_0:14;
  hence thesis by A3;
end;
