reserve V,W for Z_Module;

theorem RL5Th29:
  for V being finite-rank free Z_Module, W being Submodule of V holds
  rank W <= rank V
  proof
    let V be finite-rank free Z_Module, W be Submodule of V;
    consider I be finite Subset of V such that
    A1: I is Basis of V by ZMODUL03:def 3;
    W is finite-rank;
    then consider A be finite Subset of W such that
A2: A is Basis of W;
    reconsider AA = A as linearly-independent Subset of V
      by A2,ZMODUL03:15,VECTSP_7:def 3;
    card AA c= card I by A1,ZMODUL04:20;
    then card A c< card I or card A = card I;
    then card (card A) < card (card I) or card A = card I by CARD_2:48;
    then card A <= rank V by A1,ZMODUL03:def 5;
    hence thesis by A2,ZMODUL03:def 5;
  end;
