reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem
  for U be finite Subset of V1 st U is linearly-independent for u be
Vector of V1 st u in U for L be Linear_Combination of U\{u} holds card U=card (
  U\{u}\/{u+Sum(L)}) & U\{u}\/{u+Sum(L)} is linearly-independent
proof
  set V=V1;
  let U be finite Subset of V1 such that
A1: U is linearly-independent;
  let u be Vector of V such that
A2: u in U;
  defpred P[Nat] means for L be Linear_Combination of U\{u} st card Carrier(L)
  =$1 holds card U=card (U\{u}\/{u+Sum(L)}) & U\{u}\/{u+Sum(L)} is
  linearly-independent;
A3: for n st P[n] holds P[n+1]
  proof
    card U<>0 by A2;
    then reconsider C=card U-1 as Element of NAT by NAT_1:20;
    let n such that
A4: P[n];
    set n1=n+1;
    let L be Linear_Combination of U\{u} such that
A5: card Carrier(L)=n1;
    consider x being object such that
A6: x in Carrier L by A5,CARD_1:27,XBOOLE_0:def 1;
A7: Carrier L c= U\{u} by VECTSP_6:def 4;
    then x in U by A6,XBOOLE_0:def 5;
    then
A8: x<>0.V by A1,VECTSP_7:2;
    reconsider x as Vector of V by A6;
    x in {x} by TARSKI:def 1;
    then x in Lin({x}) by VECTSP_7:8;
    then consider X be Linear_Combination of {x} such that
A9: x=Sum(X) by VECTSP_7:7;
    set Lx=L.x;
    set LxX=Lx*X;
    Carrier LxX c= Carrier X & Carrier X c= {x} by VECTSP_6:28,def 4;
    then
A10: Carrier LxX c= {x};
    then Carrier (L-LxX) c= Carrier L \/ Carrier LxX & Carrier L \/ Carrier
    LxX c= Carrier L \/ {x} by VECTSP_6:41,XBOOLE_1:9;
    then Carrier (L-LxX) c= Carrier L \/{x};
    then
A11: Carrier (L-LxX) c= Carrier L by A6,ZFMISC_1:40;
    then Carrier (L-LxX) c= U\{u} by A7;
    then reconsider LLxX=L-LxX as Linear_Combination of U\{u} by VECTSP_6:def 4
;
A12: x in (U\{u})\/{u+Sum(LLxX)} by A6,A7,XBOOLE_0:def 3;
A13: Carrier L\{x} c= Carrier(L-LxX)
    proof
      let y be object such that
A14:  y in Carrier L\{x};
      y in Carrier L by A14,XBOOLE_0:def 5;
      then consider Y be Vector of V such that
A15:  y = Y and
A16:  L.Y <> 0.K;
      not Y in Carrier LxX by A10,A14,A15,XBOOLE_0:def 5;
      then LxX.Y =0.K;
      then (L-LxX).Y = L.Y - 0.K by VECTSP_6:40
        .= L.Y by RLVECT_1:13;
      hence thesis by A15,A16;
    end;
    X.x * x = x by A9,VECTSP_6:17
      .= 1_K*x;
    then
A17: X.x=1_K by A8,VECTSP10:4;
    (L-LxX).x = Lx - LxX.x by VECTSP_6:40
      .= Lx-Lx * 1_K by A17,VECTSP_6:def 9
      .= Lx-Lx
      .= 0.K by RLVECT_1:5;
    then not x in Carrier (L-LxX) by VECTSP_6:2;
    then Carrier (L-LxX) c= Carrier L\{x} by A11,ZFMISC_1:34;
    then
A18: Carrier (L-LxX) = Carrier L\{x} by A13;
    {x} c= Carrier L by A6,ZFMISC_1:31;
    then card Carrier (L-LxX) = n1 - card {x} by A5,A18,CARD_2:44
      .= n1-1 by CARD_1:30
      .= n;
    then
A19: (U\{u})\/{u+Sum(LLxX)} is linearly-independent by A4;
    u+Sum(LLxX) in {u+Sum(LLxX)} by TARSKI:def 1;
    then
A20: u+Sum(LLxX) in (U\{u})\/{u+Sum(LLxX)} by XBOOLE_0:def 3;
A21: not u+Sum(L) in U\{u}
    proof
      assume u+Sum(L) in U\{u};
      then
A22:  u+Sum(L) in Lin(U\{u}) by VECTSP_7:8;
A23:  (u+Sum(L))-Sum(L) = u+(Sum(L)-Sum(L)) by RLVECT_1:def 3
        .= u+0.V by RLVECT_1:5
        .= u by RLVECT_1:def 4;
      Sum (L) in Lin(U\{u}) by VECTSP_7:7;
      hence thesis by A1,A2,A22,A23,VECTSP_4:23,VECTSP_9:14;
    end;
    card U=C+1;
    then card (U\{u})= C by A2,STIRL2_1:55;
    then
A24: card ((U\{u})\/{ u+Sum L}) =C+1 by A21,CARD_2:41;
    Sum L = 0.V+Sum (L) by RLVECT_1:def 4
      .= Sum LxX +(-Sum LxX) +Sum L by RLVECT_1:5
      .= Sum LxX +(Sum L-Sum LxX)by RLVECT_1:def 3
      .= Sum LxX + Sum LLxX by VECTSP_6:47
      .= Lx*x + Sum LLxX by A9,VECTSP_6:45;
    then
A25: u+Sum(LLxX)+Lx*x = u +Sum L by RLVECT_1:def 3;
A26: not u+Sum(LLxX) in U\{u}
    proof
      assume u+Sum(LLxX) in U\{u};
      then
A27:  u+Sum(LLxX) in Lin(U\{u}) by VECTSP_7:8;
A28:  (u+Sum(LLxX))-Sum(LLxX) = u+(Sum(LLxX)-Sum(LLxX)) by RLVECT_1:def 3
        .= u+0.V by RLVECT_1:5
        .= u by RLVECT_1:def 4;
      Sum (LLxX) in Lin(U\{u}) by VECTSP_7:7;
      hence thesis by A1,A2,A27,A28,VECTSP_4:23,VECTSP_9:14;
    end;
    then
A29: ((U\{u})\/{u+Sum(LLxX)}) \ {u+Sum(LLxX)} = U\{u} by ZFMISC_1:117;
    u+Sum(LLxX)<>x by A6,A7,A26;
    hence thesis by A19,A25,A29,A20,A12,A24,MATRIX13:115;
  end;
  let L be Linear_Combination of U\{u};
A30: P[0]
  proof
    let L be Linear_Combination of U\{u};
    assume card Carrier(L)=0;
    then Carrier L = {};
    then u+Sum L = u+0.V by VECTSP_6:19
      .= u by RLVECT_1:def 4;
    hence thesis by A1,A2,ZFMISC_1:116;
  end;
  for n holds P[n] from NAT_1:sch 2(A30,A3);
  then P[card Carrier L];
  hence thesis;
end;
