 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem
  for A1, A2 being Subset of V
  st A1 is linearly-independent & A2 is linearly-independent &
  A1 /\ A2 = {} & A1 \/ A2 is linearly-dependent holds
  Lin(A1) /\ Lin(A2) <> (0).V
  proof
    let A1, A2 be Subset of V such that
    A1: A1 is linearly-independent & A2 is linearly-independent and
    A2: A1 /\ A2 = {} & A1 \/ A2 is linearly-dependent;
    consider l be Linear_Combination of A1 \/ A2 such that
    A3: Sum(l) = 0.V & Carrier(l) <> {} by A2;
    consider l1 be Linear_Combination of A1,
    l2 be Linear_Combination of A2 such that
    A4: l = l1 + l2 by A2,ZMODUL04:26;
    A5: Sum(l) = Sum(l1) + Sum(l2) by ZMODUL02:52,A4;
    A6: Carrier(l) c= Carrier(l1) \/ Carrier(l2) by A4,ZMODUL02:26;
    per cases by A3,A6;
    suppose Carrier(l1) <> {}; then
      B2: Sum(l1) <> 0.V by A1;
      B3: Sum(l1) = -Sum(l2) by A3,A5,RLVECT_1:6;
      B4: -Sum(l2) in Lin(A2) by ZMODUL01:38,ZMODUL02:64;
      B5: Sum(l1) in Lin(A1) by ZMODUL02:64;
      assume Lin(A1) /\ Lin(A2) = (0).V;
      hence contradiction by B2,B5,ZMODUL02:66,B3,B4,ZMODUL01:94;
    end;
    suppose Carrier(l2) <> {}; then
      B2: Sum(l2) <> 0.V by A1;
      B3: Sum(l2) = -Sum(l1) by A3,A5,RLVECT_1:6;
      B4: -Sum(l1) in Lin(A1) by ZMODUL01:38,ZMODUL02:64;
      B5: Sum(l2) in Lin(A2) by ZMODUL02:64;
      assume Lin(A1) /\ Lin(A2) = (0).V;
      hence contradiction by B2,B5,ZMODUL02:66,B3,B4,ZMODUL01:94;
    end;
  end;
