reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;

theorem Th24:
  for B being B_Lattice, a being Element of B holds
  X is_greater_than a iff {b` where b is Element of B: b in X} is_less_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_greater_than a implies Y is_less_than a`
  proof
    assume
A1: for b being Element of B st b in X holds a [= b;
    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 b [= a`;
  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;
