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

theorem FRds1:
  for V being Z_Module, W1, W2 being free Subspace of V,
  I1 being Basis of W1, I2 being Basis of W2 st V is_the_direct_sum_of W1,W2
  holds I1 /\ I2 = {}
  proof
    let V be Z_Module, W1, W2 be free Subspace of V,
    I1 be Basis of W1, I2 be Basis of W2 such that
    A1: V is_the_direct_sum_of W1,W2;
    assume I1 /\ I2 <> {};
    then consider v be object such that
    A2: v in I1 /\ I2 by XBOOLE_0:7;
    A3: v in I1 by A2,XBOOLE_0:def 4;
    not 0.W1 in I1 by ZMODUL02:57,VECTSP_7:def 3;
    then A4: v <> 0.V by A3,ZMODUL01:26;
    A5: v in W1 by A3;
    v in W2 by A2; then
W1: v in W1 /\ W2 by A5,VECTSP_5:3;
    W1 /\ W2 = (0).V by A1,VECTSP_5:def 4;
    then v in (0).V by W1;
    hence contradiction by A4,ZMODUL02:66;
  end;
