
theorem
  for L being Z_Lattice, L1 being Z_Sublattice of L,
  v being Vector of L, a being Element of INT.Ring holds
  v in L1 implies a * v in L1
  proof
    let L be Z_Lattice, L1 be Z_Sublattice of L,
    v be Vector of L, a be Element of INT.Ring;
    assume AS: v in L1;
    L1 is Submodule of L by ThSL1;
    hence thesis by AS,ZMODUL01:37;
  end;
