reserve V,W for Z_Module;

theorem RL5Th30:
  for V being Z_Module, A being finite linearly-independent Subset of V holds
  card A = rank Lin(A)
  proof
    let V be Z_Module, A be finite linearly-independent Subset of V;
    set W = Lin(A);
    now
      let x be object;
      assume x in A;
      then x in W by ZMODUL02:65;
      hence x in the carrier of W;
    end;
    then reconsider B = A as finite linearly-independent Subset of W
      by ZMODUL03:16,TARSKI:def 3;
    W = Lin B by ZMODUL03:20;
    then reconsider B as Basis of W by VECTSP_7:def 3;
    card B = rank W by ZMODUL03:def 5;
    hence thesis;
  end;
