theorem
  for L1,L2 being non empty LattStr st [:L1,L2:] is Lattice holds L1 is
  Lattice & L2 is Lattice
proof
  let L1,L2 be non empty LattStr such that
A1: [:L1,L2:] is Lattice;
A2: join(L1) is associative by A1,Th23;
A3: meet(L2) is associative by A1,Th23;
A4: meet(L2) is commutative by A1,Th22;
  reconsider LL = LattStr (#[:the carrier of L1, the carrier of L2:], |:join(
    L1), join(L2):|, |:meet(L1),meet(L2):|#) as non empty LattStr;
A5: join(LL) absorbs meet(LL) by A1,LATTICE2:26;
  then
A6: join(L1) absorbs meet(L1) by Th30;
A7: join(L2) is associative by A1,Th23;
A8: join(L2) is commutative by A1,Th22;
A9: meet(L1) is associative by A1,Th23;
A10: meet(L1) is commutative by A1,Th22;
A11: meet(LL) absorbs join(LL) by A1,LATTICE2:27;
  then
A12: meet(L1) absorbs join(L1) by Th30;
A13: meet(L2) absorbs join(L2) by A11,Th30;
A14: join(L2) absorbs meet(L2) by A5,Th30;
  join(L1) is commutative by A1,Th22;
  hence thesis by A2,A10,A9,A6,A12,A8,A7,A4,A3,A14,A13,LATTICE2:11;
end;
