reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LmRankSX1:
  for V being finite-rank free Z_Module,
  A being linearly-independent Subset of V holds
  ex I being finite linearly-independent Subset of V
  st A c= I & rank(V) = card(I)
  proof
    let V be finite-rank free Z_Module,
    A be linearly-independent Subset of V;
    consider I be finite linearly-independent Subset of V,
    a be Element of INT.Ring such that
    A1: a <> 0 & A c= I & a (*) V is Submodule of Lin(I) by LmRankSX11;
    take I;
    A3: V is finite-rank free Submodule of V by ZMODUL01:32;
    rank (a (*) V) <= rank Lin(I) by A1,ZMODUL05:2;
    then
    A2: rank (V) <= rank Lin(I) by A1,A3,ZMODUL06:52;
    rank (Lin(I)) <= rank (V) by ZMODUL05:2;
    then rank (Lin(I)) = rank (V) by A2,XXREAL_0:1;
    hence thesis by A1,ZMODUL05:3;
  end;
