
theorem
  for L being with_infima antisymmetric RelStr for D being Subset of L,
  x being Element of L holds {x} "/\" D is_<=_than x
proof
  let L be with_infima antisymmetric RelStr, D be Subset of L, x be Element of
  L;
  let b be Element of L;
A1: {x} "/\" D = { x "/\" h where h is Element of L : h in D } by Th42;
  assume b in {x} "/\" D;
  then consider h being Element of L such that
A2: b = x "/\" h and
  h in D by A1;
  ex w being Element of L st x >= w & h >= w & for c being Element of L st
  x >= c & h >= c holds w >= c by LATTICE3:def 11;
  hence thesis by A2,LATTICE3:def 14;
end;
