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
  for L being Lattice, p,q,r being Element of L holds
  p is_less_than {q,r} iff p [= q"/\"r
proof
  let L be Lattice, p,q,r be Element of L;
A1: q in {q,r} by TARSKI:def 2;
A2: r in {q,r} by TARSKI:def 2;
  thus p is_less_than {q,r} implies p [= q"/\"r
  proof
    assume
A3: p is_less_than {q,r};
    then
A4: p [= q by A1;
    p [= r by A2,A3;
    hence thesis by A4,FILTER_0:7;
  end;
  assume
A5: p [= q"/\"r;
  let a be Element of L;
  assume a in {q,r};
  then
A6: a = q or a = r by TARSKI:def 2;
A7: q"/\"r [= q by LATTICES:6;
  r"/\"q [= r by LATTICES:6;
  hence thesis by A5,A6,A7,LATTICES:7;
end;
