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

theorem FRdsX:
  for V being Z_Module, W1, W2 being free Subspace of V
  st V is_the_direct_sum_of W1,W2
  holds V is free
  proof
    let V be Z_Module, W1, W2 be free Subspace of V such that
    A1: V is_the_direct_sum_of W1,W2;
    set I1 = the Basis of W1;
    set I2 = the Basis of W2;
    set I = I1 \/ I2;
    the carrier of W1 c= the carrier of V by VECTSP_4:def 2;
    then A3: I1 is Subset of V by XBOOLE_1:1;
    the carrier of W2 c= the carrier of V by VECTSP_4:def 2;
    then I2 is Subset of V by XBOOLE_1:1;
    then reconsider I as Subset of V by A3,XBOOLE_1:8;
    the ModuleStr of V = Lin(I) by A1,FRds2; then
    I is base by VECTSP_7:def 3,FRds3,A1;
    hence thesis by VECTSP_7:def 4;
  end;
