
theorem
  for V being Z_Module, I, I1 being linearly-independent Subset of V
  st I1 c= I holds
  Lin(I \ I1) /\ Lin(I1) = (0).V
  proof
    let V be Z_Module, I, I1 be linearly-independent Subset of V such that
    A1: I1 c= I;
    assume Lin(I \ I1) /\ Lin(I1) <> (0).V;
    then consider v be Vector of V such that
    A2: v in Lin(I \ I1) /\ Lin(I1) & v <> 0.V by ZMODUL04:24;
    v in Lin(I \ I1) by A2,ZMODUL01:94;
    then consider l1 be Linear_Combination of I \ I1 such that
    A3: v = Sum(l1) by ZMODUL02:64;
    A4: Carrier(l1) c= I \ I1 by VECTSP_6:def 4;
    v in Lin(I1) by A2,ZMODUL01:94;
    then consider l2 be Linear_Combination of I1 such that
    A5: v = Sum(l2) by ZMODUL02:64;
    A6: Carrier(l2) c= I1 by VECTSP_6:def 4;
    reconsider ll1 = l1 as Linear_Combination of I
      by XBOOLE_1:36,ZMODUL02:10;
    reconsider ll2 = l2 as Linear_Combination of I by A1,ZMODUL02:10;
    Carrier(l1) misses Carrier(l2) by A4,A6,XBOOLE_1:64,79;
    then Carrier(ll1) /\ Carrier(ll2) = {} by XBOOLE_0:def 7;
    then A10: Carrier(ll1) /\ Carrier(-ll2) = {} by ZMODUL02:37;
    reconsider ll2x = -ll2 as Linear_Combination of I by ZMODUL02:38;
A7: Carrier(ll1 - ll2) = Carrier(ll1) \/ Carrier(ll2x)
      by A10,ZMODUL04:25;
    A8: Sum(ll1) - Sum(ll2) = 0.V by A3,A5,RLVECT_1:5;
    reconsider l = ll1 - ll2 as Linear_Combination of I by ZMODUL02:41;
    A9: Sum(l) = 0.V by A8,ZMODUL02:55;
    Carrier(ll1) <> {} by A2,A3,ZMODUL02:23;
    hence contradiction by A9,A7,VECTSP_7:def 1;
  end;
