 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 ThLin8:
  for V being Z_Module, W being free Subspace of V,
      I being Subset of V, v being Vector of V
  st I is linearly-independent & Lin(I) = (Omega).W & v in I holds
  (Omega).W = Lin(I \ {v}) + Lin{v} & Lin(I \ {v}) /\ Lin{v} = (0).V &
  Lin(I \ {v}) is free & Lin{v} is free & v <> 0.V
  proof
    let V be Z_Module, W be free Subspace of V, I be Subset of V,
        v be Vector of V such that
    A1: I is linearly-independent & Lin(I) = (Omega).W & v in I;
    A2: (I \ {v}) \/ {v} = I \/ {v} by XBOOLE_1:39
    .= I by A1,ZFMISC_1:40;
    A3: I \ {v} is linearly-independent by A1,ZMODUL02:56,XBOOLE_1:36;
    A4: v is non torsion by A1,ThLIV1;
    Lin(I \ {v}) /\ Lin{v} = (0).V
    proof
      assume Lin(I \ {v}) /\ Lin{v} <> (0).V;
      then consider x be Vector of V such that
      B2: x in Lin(I \ {v}) /\ Lin{v} & x <> 0.V by ZMODUL04:24;
      x in Lin(I \ {v}) by B2,ZMODUL01:94;
      then consider lx1 be Linear_Combination of I \ {v} such that
      B3: x = Sum(lx1) by ZMODUL02:64;
      B4: Carrier(lx1) <> {} by B2,B3,ZMODUL02:23;
      reconsider llx1 = lx1 as Linear_Combination of I
      by XBOOLE_1:36,ZMODUL02:10;
      x in Lin{v} by B2,ZMODUL01:94;
      then consider lx2 be Linear_Combination of {v} such that
      B5: -x = Sum(lx2) by ZMODUL01:38,ZMODUL02:64;
      reconsider llx2 = lx2 as Linear_Combination of I
      by A1,ZFMISC_1:31,ZMODUL02:10;
      B6: Carrier(lx1) c= I \ {v} by VECTSP_6:def 4;
      Carrier(lx2) c= {v} by VECTSP_6:def 4;
      then Carrier(lx1) misses Carrier(lx2) by B6,XBOOLE_1:64,XBOOLE_1:79;
      then Carrier(lx1) /\ Carrier(lx2) = {} by XBOOLE_0:def 7;
      then B7: Carrier(llx1 + llx2) = Carrier(llx1) \/ Carrier(llx2)
        by ZMODUL04:25;
      B8: Sum(llx1) + Sum(llx2) = 0.V by B3,B5,RLVECT_1:5;
      reconsider llx = llx1 + llx2 as Linear_Combination of I
      by ZMODUL02:27;
      Sum(llx) = 0.V by B8,ZMODUL02:52;
      hence contradiction by A1,B4,B7;
    end;
    hence thesis by A2,A3,A4,A1,ZMODUL02:72,ThnTV4;
  end;
