
theorem
  for x being object, L being Z_Lattice, L1, L2 being Z_Sublattice of L holds
  x in L1 & L1 is Z_Sublattice of L2 implies x in L2
  proof
    let x be object, L be Z_Lattice, L1, L2 be Z_Sublattice of L;
    assume AS: x in L1 & L1 is Z_Sublattice of L2;
    then A1: L1 is Submodule of L2 by ThSL1;
    L1 is Submodule of L & L2 is Submodule of L by ThSL1;
    hence thesis by AS,A1,ZMODUL01:23;
  end;
