theorem Th38:
  L1 is D_Lattice & L2 is D_Lattice iff [:L1,L2:] is D_Lattice
proof
  thus L1 is D_Lattice & L2 is D_Lattice implies [:L1,L2:] is D_Lattice
  proof
    assume that
A1: L1 is D_Lattice and
A2: L2 is D_Lattice;
A3: join(L2) is_distributive_wrt meet(L2) by A2,LATTICE2:21;
    join(L1) is_distributive_wrt meet(L1) by A1,LATTICE2:21;
    then |:join(L1),join(L2):| is_distributive_wrt |:meet(L1),meet(L2):| by A3
,Th29;
    hence thesis by LATTICE2:22;
  end;
  assume [:L1,L2:] is D_Lattice;
  then
A4: join([:L1,L2:]) is_distributive_wrt meet([:L1,L2:]) by LATTICE2:21;
  then
A5: join(L2) is_distributive_wrt meet(L2) by Th29;
  join(L1) is_distributive_wrt meet(L1) by A4,Th29;
  hence thesis by A5,LATTICE2:22;
end;
