reserve V,W for Z_Module;

theorem
  for V being finite-rank free Z_Module, W being Submodule of V,
  n being Nat holds
  n Submodules_of W c= n Submodules_of V
  proof
    let V be finite-rank free Z_Module, W be Submodule of V, n be Nat;
    let x be object;
    assume x in n Submodules_of W;
    then consider W1 being strict free Submodule of W such that
    A1: W1 = x and
    A2: rank W1 = n by RL5Def4;
    reconsider W1 as strict free Submodule of V by ZMODUL01:42;
    W1 in n Submodules_of V by A2,RL5Def4;
    hence thesis by A1;
  end;
