theorem Th42:
  L1 is 0_Lattice & L2 is 0_Lattice implies Bottom [:L1,L2:] = [
  Bottom L1, Bottom L2]
proof
  assume that
A1: L1 is 0_Lattice and
A2: L2 is 0_Lattice;
A3: now
    let a be Element of [:L1,L2:];
    consider p1,p2 such that
A4: a = [p1,p2] by DOMAIN_1:1;
    thus [Bottom L1,Bottom L2]"/\"a = [Bottom L1"/\"p1,Bottom L2"/\"p2] by A4
,Th21
      .= [Bottom L1,Bottom L2"/\" p2] by A1
      .= [Bottom L1,Bottom L2] by A2;
    hence a"/\"[Bottom L1,Bottom L2]=[Bottom L1,Bottom L2];
  end;
  [:L1,L2:] is lower-bounded by A1,A2,Th39;
  hence thesis by A3,LATTICES:def 16;
end;
