theorem Th39:
  L1 is lower-bounded & L2 is lower-bounded iff [:L1,L2:] is lower-bounded
proof
  thus L1 is lower-bounded & L2 is lower-bounded implies [:L1,L2:] is
  lower-bounded
  proof
    given p1 such that
A1: p1"/\"q1 = p1 & q1"/\"p1 = p1;
    given p2 such that
A2: p2"/\"q2 = p2 & q2"/\"p2 = p2;
    take a = [p1,p2];
    let b be Element of [:L1,L2:];
    consider q1,q2 such that
A3: b = [q1,q2] by DOMAIN_1:1;
    thus a"/\"b = [p1"/\"q1,p2"/\"q2] by A3,Th21
      .= [p1,p2"/\" q2] by A1
      .= a by A2;
    hence b"/\"a = a;
  end;
  given a being Element of [:L1,L2:] such that
A4: for b being Element of [:L1,L2:] holds a"/\"b = a & b"/\"a = a;
  consider p1,p2 such that
A5: a = [p1,p2] by DOMAIN_1:1;
  thus L1 is lower-bounded
  proof
    set q2 = the Element of L2;
    take p1;
    let q1;
    a = a"/\"[q1,q2] by A4
      .= [p1"/\"q1,p2"/\"q2] by A5,Th21;
    hence thesis by A5,XTUPLE_0:1;
  end;
  set q1 = the Element of L1;
  take p2;
  let q2;
  a = a"/\"[q1,q2] by A4
    .= [p1"/\"q1,p2"/\"q2] by A5,Th21;
  hence thesis by A5,XTUPLE_0:1;
end;
