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;
reserve p9,q9 for Element of LattPOSet L;
reserve C for complete Lattice,
  a,a9,b,b9,c,d for Element of C,
  X,Y for set;

theorem
  "\/"{a} = a & "/\"{a} = a
proof
A1: a in {a} by TARSKI:def 1;
  {a} is_less_than a
  proof
    let b;
    assume b in {a};
    hence b [= a by TARSKI:def 1;
  end;
  hence "\/"{a} = a by A1,Th40;
  a is_less_than {a}
  proof
    let b;
    assume b in {a};
    hence a [= b by TARSKI:def 1;
  end;
  hence thesis by A1,Th41;
end;
