theorem Th43:
  L1 is 1_Lattice & L2 is 1_Lattice implies Top [:L1,L2:] = [Top L1,Top L2]
proof
  assume that
A1: L1 is 1_Lattice and
A2: L2 is 1_Lattice;
A3: now
    let a be Element of [:L1,L2:];
    consider p1,p2 such that
A4: a = [p1,p2] by DOMAIN_1:1;
    thus [Top L1,Top L2]"\/"a = [Top L1"\/"p1,Top L2"\/"p2] by A4,Th21
      .= [ Top L1,Top L2"\/" p2] by A1
      .= [Top L1,Top L2] by A2;
    hence a"\/"[Top L1,Top L2] = [Top L1,Top L2];
  end;
  [:L1,L2:] is upper-bounded by A1,A2,Th40;
  hence thesis by A3,LATTICES:def 17;
end;
