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"\/"b = "\/"{a,b} & a"/\"b = "/\"{a,b}
proof
A1: {a,b} is_less_than a"\/"b
  proof
    let c;
    assume
A2: c in {a,b};
A3: a [= a"\/"b by LATTICES:5;
    b [= b"\/"a by LATTICES:5;
    hence thesis by A2,A3,TARSKI:def 2;
  end;
A4: a in {a,b} by TARSKI:def 2;
A5: b in {a,b} by TARSKI:def 2;
  now
    let c;
    assume
A6: {a,b} is_less_than c;
    then
A7: a [= c by A4;
    b [= c by A5,A6;
    hence a"\/"b [= c by A7,FILTER_0:6;
  end;
  hence a"\/"b = "\/"{a,b} by A1,Def21;
  a"/\"b is_less_than {a,b}
  proof
    let c;
    assume c in {a,b};
    then c = a or c = b & b"/\"a = a"/\"b by TARSKI:def 2;
    hence thesis by LATTICES:6;
  end;
  then
A8: a"/\"b in { c: c is_less_than {a,b}};
  { c: c is_less_than {a,b}} is_less_than a"/\"b
  proof
    let d be Element of C;
    assume d in { c: c is_less_than {a,b}};
    then
A9: ex c st d = c & c is_less_than {a,b};
    then
A10: d [= a by A4;
    d [= b by A5,A9;
    hence thesis by A10,FILTER_0:7;
  end;
  hence thesis by A8,Th40;
end;
