reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

theorem Th40:
  L1 is upper-bounded & L2 is upper-bounded iff [:L1,L2:] is upper-bounded
proof
  thus L1 is upper-bounded & L2 is upper-bounded implies [:L1,L2:] is
  upper-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 upper-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;
