reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for V being Z_Module, W being Subspace of V,
      W1, W2 being free Subspace of V st
  W1 /\ W2 = (0).V & the ModuleStr of W = W1 + W2
  holds W is free
  proof
    let V be Z_Module, W be Subspace of V,
        W1, W2 be free Subspace of V such that
A1: W1 /\ W2 = (0).V & the ModuleStr of W = W1 + W2;
    reconsider Ws = W1 + W2 as free Subspace of V by A1,ThDirectSum;
    consider I be Subset of Ws such that
a3: I is base by VECTSP_7:def 4;
A3: I is linearly-independent & Ws = Lin(I) by VECTSP_7:def 3,a3;
    reconsider IV = I as linearly-independent Subset of V
      by A3,ZMODUL03:15;
    reconsider IW = IV as linearly-independent Subset of W by A1,ZMODUL03:16;
    the ModuleStr of W = Lin(IV) by A1,A3,ZMODUL03:20
    .= Lin(IW) by ZMODUL03:20; then
    IW is base by VECTSP_7:def 3;
    hence thesis by VECTSP_7:def 4;
  end;
