theorem
  for B being B_Lattice, a being Element of B holds
  X is_less_than a iff {b` where b is Element of B: b in X} is_greater_than a`
proof
  let B be B_Lattice, a be Element of B;
  set Y = {b` where b is Element of B: b in X};
  thus X is_less_than a implies Y is_greater_than a`
  proof
    assume
A1: for b being Element of B st b in X holds b [= a;
    let b be Element of B;
    assume b in Y;
    then ex c being Element of B st ( b = c`)&( c in X);
    hence thesis by A1,LATTICES:26;
  end;
  assume
A2: for b being Element of B st b in Y holds a` [= b;
  let b be Element of B;
  assume b in X;
  then
A3: b` in Y;
A4: a`` = a;
  b`` = b;
  hence thesis by A2,A3,A4,LATTICES:26;
end;
