reserve V,W for Z_Module;

theorem
  for V being finite-rank free Z_Module, n being Nat holds
  rank V < n implies n Submodules_of V = {}
  proof
    let V be finite-rank free Z_Module, n be Nat;
    assume that
    A1: rank V < n and
    A2: n Submodules_of V <> {};
    consider x being object such that
    A3: x in n Submodules_of V by A2,XBOOLE_0:def 1;
    ex W being strict free Submodule of V st W = x & rank W = n by A3,RL5Def4;
    hence contradiction by A1,RL5Th29;
  end;
