theorem Th29:
  for V1 be finite-dimensional VectSp of K for F be nilpotent
linear-transformation of V1,V1 ex J be non-empty FinSequence_of_Jordan_block of
  0.K,K, b1 be OrdBasis of V1 st AutMt(F,b1,b1) = block_diagonal(J,0.K)
proof
  defpred P[Nat] means for V1 be finite-dimensional VectSp of K for F be
  nilpotent linear-transformation of V1,V1 st deg F=$1 holds ex J be
FinSequence_of_Jordan_block of 0.K,K, b1 be OrdBasis of V1 st AutMt(F,b1,b1) =
  block_diagonal(J,0.K) & for i st i in dom J holds (Len J).i <> 0;
  let V1 be finite-dimensional VectSp of K;
  let F be nilpotent linear-transformation of V1,V1;
A1: for n st P[n] holds P[n+1]
  proof
    let n such that
A2: P[n];
    let V1 be finite-dimensional VectSp of K;
    set n1=n+1;
    let F be nilpotent linear-transformation of V1,V1 such that
A3: deg F=n1;
    set BAS = the Basis of V1;
A4: BAS is linearly-independent by VECTSP_7:def 3;
A5: Lin(BAS) = the ModuleStr of V1 by VECTSP_7:def 3;
    set IM=im F|^1;
    reconsider FI=F|IM as linear-transformation of IM,IM by VECTSP11:32;
    reconsider FI as nilpotent linear-transformation of IM,IM by Th17;
    deg FI+1=n1 by A3,Th18,NAT_1:11;
    then consider
    J be FinSequence_of_Jordan_block of 0.K,K, b1 be OrdBasis of IM
    such that
A6: AutMt(FI,b1,b1) = block_diagonal(J,0.K) and
A7: for i st i in dom J holds (Len J).i <> 0 by A2;
A8: len b1 = len AutMt(FI,b1,b1) by MATRIX_0:def 2
      .= Sum Len J by A6,MATRIXJ1:def 5;
    then
A9: dom b1= Seg Sum Len J by FINSEQ_1:def 3;
    set L=len J;
    set LJ=Len J;
    set S=Sum LJ;
    defpred Q[Nat,Nat] means $2 in dom LJ & $2 <= $1 & Sum(LJ| ($2-'1)) <= $1-'
    $2;
    defpred R[object,object] means
    for i,k st i=$1 & k=$2 holds Q[i,k] & i-'k <= Sum
    (LJ|k) & for n st Q[i,n] holds n<=k;
A10: for x being object st x in Seg(S+L)
   ex y being object st y in NAT & R[x,y]
    proof
      let x be object such that
A11:  x in Seg (S+L);
      reconsider i=x as Nat by A11;
      L<>0
      proof
        assume
A12:    L=0;
        then LJ=<*>NAT;
        hence thesis by A11,A12,RVSUM_1:72;
      end;
      then
A13:  1<=L by NAT_1:14;
      1-'1=0 by XREAL_1:232;
      then
A14:  Sum(LJ| (1-'1))=0 by RVSUM_1:72;
      defpred q[Nat] means $1 in dom LJ & $1 <= i & Sum(LJ| ($1-'1)) <= i-'$1;
A15:  for k st q[k] holds k <= L
      proof
        let k;
        assume q[k];
        then k<=len LJ by FINSEQ_3:25;
        hence thesis by CARD_1:def 7;
      end;
      len LJ=L by CARD_1:def 7;
      then
A16:  0<=i-'1 & 1 in dom LJ by A13,FINSEQ_3:25;
      1<=i by A11,FINSEQ_1:1;
      then
A17:  ex k st q[k] by A14,A16;
      consider k such that
A18:  q[k] and
A19:  for n st q[n] holds n <= k from NAT_1:sch 6(A15,A17);
A20:  i-'k <= Sum (LJ|k)
      proof
        assume
A21:    i-'k > Sum (LJ|k);
        then i-'k >= Sum (LJ|k)+1 by NAT_1:13;
        then
A22:    i-'k-1>=Sum (LJ|k)+1-1 by XREAL_1:9;
A23:    i-'k=i-k by A18,XREAL_1:233;
A24:    k+1<=len LJ
        proof
          assume k+1>len LJ;
          then
A25:      k>=len LJ by NAT_1:13;
          then i-k>S by A21,A23,FINSEQ_1:58;
          then
A26:      i-k+k>S+k by XREAL_1:6;
          len LJ=L by CARD_1:def 7;
          then S+k>=S+L by A25,XREAL_1:6;
          then i> S+L by A26,XXREAL_0:2;
          hence thesis by A11,FINSEQ_1:1;
        end;
        1<=k+1 by NAT_1:14;
        then
A27:    k+1 in dom LJ by A24,FINSEQ_3:25;
        i-'k>=1 by A21,NAT_1:14;
        then
A28:    i-k+k >=1+k by A23,XREAL_1:6;
        then i-'(k+1)=i-(k+1) by XREAL_1:233;
        then Sum(LJ| (k+1-'1)) <= i-'(k+1) by A22,A23,NAT_D:34;
        then k+1<=k by A19,A28,A27;
        hence thesis by NAT_1:13;
      end;
      take k;
      thus k in NAT by ORDINAL1:def 12;
      let i9,k9 be Nat;
      assume i9=x & k9=k;
      hence thesis by A18,A19,A20;
    end;
    consider r be Function of Seg (S+L),NAT such that
A29: for x being object st x in Seg (S+L) holds R[x,r.x]
        from FUNCT_2:sch 1(A10);
    defpred P[object,object] means
     for i,k st i=$1 & k=r.i holds (i -' k = Sum (LJ| (
k-'1)) implies (F.$2 = b1.(i -' k+1) & i-'k+1 in dom b1)) & (i -' k <> Sum (LJ|
(k-'1)) implies ($2 = b1.(i -' k) & i-'k in dom b1 & min(LJ,i-'k)=k & ((i-'k <
Sum (LJ|k) implies F.$2 = b1.(i -' k+1) & i -' k+1 in dom b1) & (i-'k = Sum (LJ
    |k) implies F.$2 = 0.V1))));
A30: dom r=Seg (S+L) by FUNCT_2:def 1;
A31: FI=Mx2Tran(AutMt(FI,b1,b1),b1,b1) by MATRLIN2:34;
A32: for x being object st x in Seg (S+L)
ex y being object st y in the carrier of V1 & P[x,y]
    proof
      let x be object such that
A33:  x in Seg (S+L);
      reconsider i=x as Nat by A33;
      r.i=r/.i by A30,A33,PARTFUN1:def 6;
      then reconsider k=r.i as Element of NAT;
A34:  i-'k <= Sum (LJ|k) by A29,A33;
A35:  Q[i,k] by A29,A33;
      then
A36:  LJ.k =len (J.k) by MATRIXJ1:def 3;
      k<=len LJ by A35,FINSEQ_3:25;
      then
A37:  Sum (LJ|k) <= Sum (LJ| (len LJ)) by POLYNOM3:18;
      1<=k by A35,FINSEQ_3:25;
      then
A38:  k-'1=k-1 by XREAL_1:233;
      then k=k-'1+1;
      then LJ|k=(LJ| (k-'1))^<*LJ.k*> by A35,FINSEQ_5:10;
      then
A39:  dom LJ=dom J & Sum (LJ|k)=Sum (LJ| (k-'1)) + len (J.k) by A36,
MATRIXJ1:def 3,RVSUM_1:74;
A40:  LJ| (len LJ) =LJ by FINSEQ_1:58;
      per cases;
      suppose
A41:    i -' k = Sum (LJ| (k-'1));
        b1/.(i-'k+1) in IM & b1/.(i-'k+1) is Element of V1 by VECTSP_4:10;
        then consider y be Element of V1 such that
A42:    (F|^1).y=b1/.(i-'k+1) by RANKNULL:13;
        take y;
        thus y in the carrier of V1;
        i-'k <> Sum (LJ|k) by A7,A35,A36,A39,A41;
        then i-'k < Sum (LJ|k) by A34,XXREAL_0:1;
        then i-'k+1 <=Sum(LJ|k) by NAT_1:13;
        then
A43:    1<= i-'k+1 & i-'k+1 <= S by A37,A40,NAT_1:11,XXREAL_0:2;
        then i-'k+1 in dom b1 by A9;
        then b1/.(i-'k+1) = b1.(i-'k+1) by PARTFUN1:def 6;
        hence thesis by A9,A41,A43,A42,VECTSP11:19;
      end;
      suppose
A44:    i -' k <> Sum (LJ| (k-'1));
        take y=b1/.(i-'k);
        y in IM;
        then y in V1 by VECTSP_4:9;
        hence y in the carrier of V1;
        i -' k>Sum (LJ| (k-'1)) by A35,A44,XXREAL_0:1;
        then
A45:    1<= i-'k by NAT_1:14;
        i-'k <= S by A34,A37,A40,XXREAL_0:2;
        then
A46:    i-'k in dom b1 by A9,A45;
        i-'k <= Sum (LJ|k) by A29,A33;
        then
A47:    min(LJ,i-'k)<=k by A9,A46,MATRIXJ1:def 1;
A48:    min(LJ,i-'k)=k
        proof
          assume min(LJ,i-'k)<>k;
          then min(LJ,i-'k)<k-'1+1 by A38,A47,XXREAL_0:1;
          then min(LJ,i-'k) <=k-'1 by NAT_1:13;
          then
A49:      Sum(LJ|min(LJ,i-'k)) <= Sum (LJ| (k-'1)) by POLYNOM3:18;
          i-'k <= Sum(LJ|min(LJ,i-'k)) by A9,A46,MATRIXJ1:def 1;
          then i-'k <= Sum (LJ| (k-'1)) by A49,XXREAL_0:2;
          hence thesis by A35,A44,XXREAL_0:1;
        end;
A50:    Len J|k=LJ|k by MATRIXJ1:17;
A51:    now
          assume
A52:      i-'k = Sum (LJ|k);
          F.(b1/.(i-'k)) = FI.(b1/.(i-'k)) by FUNCT_1:49
            .= 0.K*(b1/.(i-'k)) by A6,A31,A46,A48,A50,A52,Th24
            .= 0.IM by VECTSP_1:14
            .= 0.V1 by VECTSP_4:11;
          hence F.y = 0.V1;
        end;
A53:    now
          assume
A54:      i-'k < Sum (LJ|k);
          then i-'k+1<=Sum (LJ|k) by NAT_1:13;
          then
A55:      1<=i-'k+1 & i-'k+1<=S by A37,A40,NAT_1:11,XXREAL_0:2;
          then
A56:      i-'k+1 in dom b1 by A9;
          F.(b1/.(i-'k)) = FI.(b1/.(i-'k)) by FUNCT_1:49
            .= 0.K*(b1/.(i-'k))+(b1/.(i-'k+1)) by A6,A31,A46,A48,A50,A54,Th24
            .= 0.IM+(b1/.(i-'k+1)) by VECTSP_1:14
            .= b1/.(i-'k+1) by RLVECT_1:def 4
            .= b1.(i-'k+1) by A56,PARTFUN1:def 6;
          hence F.y = b1.(i -' k+1) & i -' k+1 in dom b1 by A9,A55;
        end;
        let i9,k9 be Nat;
        assume x=i9 & k9=r.i9;
        hence thesis by A44,A46,A48,A53,A51,PARTFUN1:def 6;
      end;
    end;
    consider B be Function of Seg (S+L),the carrier of V1 such that
A57: for x being object st x in Seg (S+L) holds P[x,B.x]
from FUNCT_2:sch 1(A32);
A58: rng B c= the carrier of V1 by RELAT_1:def 19;
A59: dom B=Seg (S+L) by FUNCT_2:def 1;
    then reconsider B as FinSequence by FINSEQ_1:def 2;
    reconsider B as FinSequence of V1 by A58,FINSEQ_1:def 4;
    reconsider RNG=rng B as Subset of V1 by FINSEQ_1:def 4;
    now
      rng b1 is Basis of IM by MATRLIN:def 2;
      then rng b1 is linearly-independent Subset of IM by VECTSP_7:def 3;
      then reconsider rngb1=rng b1 as linearly-independent Subset of V1 by
VECTSP_9:11;
      set RB={v1 where v1 is Vector of V1: ex i,k st i in Seg(L+S) & k=r.i &
      v1=B.i & i-'k <> Sum (LJ| (k-'1)) & i-'k = Sum (LJ|k)};
      set RA={v1 where v1 is Vector of V1: ex i,k st i in Seg(L+S) & k=r.i &
      v1=B.i & i-'k < Sum (LJ|k)};
A60:  RA c= the carrier of V1
      proof
        let x be object;
        assume x in RA;
        then ex v1 be Vector of V1 st x=v1 & ex i,k st i in Seg(L+S) & k=r.i
        & v1=B.i & i-'k < Sum (LJ|k);
        hence thesis;
      end;
      RB c= the carrier of V1
      proof
        let x be object;
        assume x in RB;
        then ex v1 be Vector of V1 st x=v1 & ex i,k st i in Seg(L+S) & k=r.i
        & v1=B.i & i-'k <> Sum (LJ| (k-'1)) & i-'k = Sum (LJ|k);
        hence thesis;
      end;
      then reconsider RA,RB as Subset of V1 by A60;
      let l be Linear_Combination of RNG such that
A61:  Sum l = 0.V1;
A62:  F|RA is one-to-one
      proof
        let x1,x2 be object such that
A63:    x1 in dom (F|RA) and
A64:    x2 in dom (F|RA) and
A65:    (F|RA).x1=(F|RA).x2;
A66:    (F|RA).x1=F.x1 & (F|RA).x2=F.x2 by A63,A64,FUNCT_1:47;
A67:    dom(F|RA)=dom F /\ RA by RELAT_1:61;
        then x1 in RA by A63,XBOOLE_0:def 4;
        then consider v1 be Vector of V1 such that
A68:    x1=v1 and
A69:    ex i1,k1 be Nat st i1 in Seg(L+S) & k1=r.i1 & v1=B.i1 & i1-'
        k1 < Sum (LJ|k1);
        consider i1,k1 be Nat such that
A70:    i1 in Seg(L+S) & k1=r.i1 and
A71:    v1=B.i1 and
A72:    i1-'k1 < Sum (LJ|k1) by A69;
        k1<=i1 by A29,A70;
        then
A73:    i1-'k1=i1-k1 by XREAL_1:233;
A74:    k1 in dom LJ by A29,A70;
        then 1<=k1 by FINSEQ_3:25;
        then
A75:    k1-'1=k1-1 by XREAL_1:233;
        then k1-'1+1<=len LJ by A74,FINSEQ_3:25;
        then
A76:    k1-'1 <=len LJ by NAT_1:13;
A77:    b1 is one-to-one by MATRLIN:def 2;
A78:    dom LJ=dom J by MATRIXJ1:def 3;
        then
A79:    k1-'1 in dom LJ implies LJ.(k1-'1)<>0 by A7;
        x2 in RA by A64,A67,XBOOLE_0:def 4;
        then consider v2 be Vector of V1 such that
A80:    x2 = v2 and
A81:    ex i2,k2 be Nat st i2 in Seg(L+S) & k2=r.i2 & v2=B.i2 & i2-'
        k2 < Sum (LJ|k2);
        consider i2,k2 be Nat such that
A82:    i2 in Seg(L+S) & k2=r.i2 and
A83:    v2=B.i2 and
A84:    i2-'k2 < Sum (LJ|k2) by A81;
A85:    k2<=i2 by A29,A82;
        then
A86:    i2-'k2=i2-k2 by XREAL_1:233;
A87:    k2 in dom LJ by A29,A82;
        then 1<=k2 by FINSEQ_3:25;
        then
A88:    k2-'1=k2-1 by XREAL_1:233;
        then k2-'1+1<=len LJ by A87,FINSEQ_3:25;
        then
A89:    k2-'1 <=len LJ by NAT_1:13;
A90:    k2-'1 in dom LJ implies LJ.(k2-'1)<>0 by A7,A78;
        per cases;
        suppose
A91:      i1-'k1=Sum (LJ| (k1-'1)) & i2-'k2=Sum (LJ| (k2-'1));
          then
A92:      F.v2=b1.(i2 -' k2+1) & i2 -' k2+1 in dom b1 by A57,A82,A83;
          F.v1=b1.(i1 -' k1+1) & i1 -' k1+1 in dom b1 by A57,A70,A71,A91;
          then i1-'k1+1=i2-'k2+1 by A65,A66,A68,A80,A77,A92;
          then k1-'1=k2-'1 by A76,A89,A79,A90,A91,MATRIXJ1:11;
          then i1-k1=i2-k1 by A85,A73,A75,A88,A91,XREAL_1:233;
          hence thesis by A68,A71,A80,A83;
        end;
        suppose
A93:      i1-'k1=Sum (LJ| (k1-'1)) & i2-'k2<>Sum (LJ| (k2-'1));
          then
A94:      min(LJ,i2-'k2)=k2 by A57,A82;
A95:      F.v1=b1.(i1-'k1+1) & i1-'k1+1 in dom b1 by A57,A70,A71,A93;
          F.v2=b1.(i2-'k2+1) & i2-'k2+1 in dom b1 by A57,A82,A83,A84,A93;
          then
A96:      i1-'k1+1=i2-'k2+1 by A65,A66,A68,A80,A77,A95;
          k1-'1 <>0
          proof
            assume k1-'1=0;
            then LJ| (k1-'1)=<*>REAL;
            hence thesis by A29,A82,A93,A96,RVSUM_1:72;
          end;
          then k1-'1 >=1 by NAT_1:14;
          then
A97:      k1-'1 in dom LJ by A76,FINSEQ_3:25;
          then LJ.(k1-'1)<>0 by A7,A78;
          hence thesis by A84,A93,A94,A96,A97,MATRIXJ1:6;
        end;
        suppose
A98:      i1-'k1<>Sum (LJ| (k1-'1)) & i2-'k2=Sum (LJ| (k2-'1));
          then
A99:      min(LJ,i1-'k1)=k1 by A57,A70;
A100:     F.v2=b1.(i2-'k2+1) & i2-'k2+1 in dom b1 by A57,A82,A83,A98;
          F.v1=b1.(i1-'k1+1) & i1-'k1+1 in dom b1 by A57,A70,A71,A72,A98;
          then
A101:     i1-'k1+1=i2-'k2+1 by A65,A66,A68,A80,A77,A100;
          k2-'1 <>0
          proof
            assume k2-'1=0;
            then i1-'k1 =0 by A98,A101,RVSUM_1:72;
            hence thesis by A29,A70,A98;
          end;
          then k2-'1 >=1 by NAT_1:14;
          then
A102:     k2-'1 in dom LJ by A89,FINSEQ_3:25;
          then LJ.(k2-'1)<>0 by A7,A78;
          hence thesis by A72,A98,A99,A101,A102,MATRIXJ1:6;
        end;
        suppose
A103:     i1-'k1<>Sum (LJ| (k1-'1)) & i2-'k2<>Sum (LJ| (k2-'1));
          then
A104:     min(LJ,i2-'k2)=k2 by A57,A82;
A105:     F.v2=b1.(i2-'k2+1) & i2-'k2+1 in dom b1 by A57,A82,A83,A84,A103;
          F.v1=b1.(i1-'k1+1) & i1-'k1+1 in dom b1 by A57,A70,A71,A72,A103;
          then i1-'k1+1=i2-'k2+1 by A65,A66,A68,A80,A77,A105;
          then i1 - k1 =i2-k1 by A57,A70,A73,A86,A103,A104;
          hence thesis by A68,A71,A80,A83;
        end;
      end;
A106: RB c= rngb1
      proof
        let x be object;
        assume x in RB;
        then consider v1 be Vector of V1 such that
A107:   x=v1 and
A108:   ex i,k st i in Seg(L+S) & k=r.i & v1=B.i & i-'k <> Sum (LJ| (
        k-'1)) & i-'k = Sum (LJ|k);
        consider i,k such that
A109:   i in Seg(L+S) & k=r.i & v1=B.i & i-'k <> Sum (LJ| (k-'1)) and
        i-'k = Sum (LJ|k) by A108;
        v1= b1.(i -' k) & i-'k in dom b1 by A57,A109;
        hence thesis by A107,FUNCT_1:def 3;
      end;
A110: Carrier l c= RB\/RA
      proof
        let x be object such that
A111:   x in Carrier l;
        reconsider v1=x as Vector of V1 by A111;
        Carrier l c= RNG by VECTSP_6:def 4;
        then consider i be object such that
A112:   i in dom B and
A113:   B.i=v1 by A111,FUNCT_1:def 3;
        reconsider i as Nat by A112;
        r.i=r/.i by A30,A59,A112,PARTFUN1:def 6;
        then reconsider k=r.i as Element of NAT;
A114:   i-'k <= Sum (LJ|k) by A29,A59,A112;
        per cases by A114,XXREAL_0:1;
        suppose
A115:     i-'k = Sum (LJ|k);
A116:     Q[i,k] by A29,A59,A112;
          then 1<=k by FINSEQ_3:25;
          then k-'1=k-1 by XREAL_1:233;
          then k-'1+1=k;
          then LJ|k=(LJ| (k-'1))^<*LJ.k*> by A116,FINSEQ_5:10;
          then dom LJ=dom J & i-'k = Sum (LJ| (k-'1))+LJ.k by A115,
MATRIXJ1:def 3,RVSUM_1:74;
          then i-'k <>Sum (LJ| (k-'1)) by A7,A116;
          then v1 in RB or v1 in RA by A59,A112,A113,A115;
          hence thesis by XBOOLE_0:def 3;
        end;
        suppose
          i-'k < Sum (LJ|k);
          then v1 in RB or v1 in RA by A59,A112,A113;
          hence thesis by XBOOLE_0:def 3;
        end;
      end;
      F.:RA c= rngb1
      proof
        let y be object;
        assume y in F.:RA;
        then consider x being object such that
        x in dom F and
A117:   x in RA and
A118:   y=F.x by FUNCT_1:def 6;
        consider v1 be Vector of V1 such that
A119:   x=v1 and
A120:   ex i,k st i in Seg(L+S) & k=r.i & v1=B.i&i-'k < Sum (LJ|k) by A117;
        consider i,k such that
A121:   i in Seg(L+S) & k=r.i & v1=B.i & i-'k < Sum (LJ|k) by A120;
        i-'k <> Sum (LJ| (k-'1)) or i-'k = Sum (LJ| (k-'1));
        then F.v1= b1.(i -' k+1) & i -' k+1 in dom b1 by A57,A121;
        hence thesis by A118,A119,FUNCT_1:def 3;
      end;
      then
A122: F.:RA is linearly-independent Subset of V1 by VECTSP_7:1;
      F.:RB c= {0.V1}
      proof
        let y be object;
        assume y in F.:RB;
        then consider x being object such that
        x in dom F and
A123:   x in RB and
A124:   y=F.x by FUNCT_1:def 6;
        consider v1 be Vector of V1 such that
A125:   x=v1 and
A126:   ex i,k st i in Seg(L+S) & k=r.i & v1=B.i & i-'k <> Sum (LJ| (
        k-'1)) & i-'k = Sum (LJ|k) by A123;
        F.v1= 0.V1 by A57,A126;
        hence thesis by A124,A125,TARSKI:def 1;
      end;
      then Carrier l c= RB by A61,A110,A62,A122,VECTSP11:44;
      then Carrier l c= rngb1 by A106;
      then l is Linear_Combination of rngb1 by VECTSP_6:def 4;
      hence Carrier l ={} by A61,VECTSP_7:def 1;
    end;
    then
A127: RNG is linearly-independent Subset of V1 by VECTSP_7:def 1;
    reconsider BAS,RNG as finite Subset of V1;
    consider C be finite Subset of V1 such that
    C c= BAS and
A128: card C = card BAS - card RNG and
A129: the ModuleStr of V1= Lin(RNG\/C) by A127,A5,VECTSP_9:19;
A130: (Omega).Lin(BAS) = (Omega).V1 by VECTSP_7:def 3;
    then
A131: dim V1 =dim Lin BAS by VECTSP_9:28;
    defpred W[Nat] means $1 <= card C implies ex f be FinSequence of V1 st f
is one-to-one & len f = card C & RNG misses rng f & RNG\/rng f is Basis of V1 &
    for i st i in dom f & i<= $1 holds F.(f.i)=0.V1;
A132: for n st W[n] holds W[n+1]
    proof
      let n such that
A133: W[n];
      set n1=n+1;
      assume
A134: n1<=card C;
      then consider f be FinSequence of V1 such that
A135: f is one-to-one and
A136: len f = card C and
A137: RNG misses rng f and
A138: RNG\/rng f is Basis of V1 and
A139: for i st i in dom f & i<= n holds F.(f.i)=0.V1 by A133,NAT_1:13;
      per cases;
      suppose
        F.(f.n1)=0.V1;
        then for i st i in dom f & i<=n1 holds F.(f.i)=0.V1 by A139,NAT_1:8;
        hence thesis by A135,A136,A137,A138;
      end;
      suppose
A140:   F.(f.n1)<>0.V1;
        reconsider Rf=RNG\/rng f as finite Subset of V1 by A138;
        reconsider rngB1=rng b1 as Basis of IM by MATRLIN:def 2;
        set fn=f/.n1;
        1<=n1 by NAT_1:14;
        then
A141:   n1 in dom f by A134,A136,FINSEQ_3:25;
        then
A142:   f/.n1=f.n1 by PARTFUN1:def 6;
A143:   rng b1 c= F.:RNG
        proof
A144:     dom F=[#]V1 by FUNCT_2:def 1;
          let y be object;
          assume y in rng b1;
          then consider x being object such that
A145:     x in dom b1 and
A146:     b1.x=y by FUNCT_1:def 3;
          reconsider x as Element of NAT by A145;
A147:     len LJ=L & x<=S by A8,A145,CARD_1:def 7,FINSEQ_3:25;
          set m=min(LJ,x);
A148:     x<= Sum(LJ|m) by A9,A145,MATRIXJ1:def 1;
A149:     m in dom LJ by A9,A145,MATRIXJ1:def 1;
          then m<=len LJ by FINSEQ_3:25;
          then m+x<=L+S by A147,XREAL_1:7;
          then
A150:     m+x-1<=L+S-1 by XREAL_1:9;
          set x1=x-'1;
A151:     1<=x by A145,FINSEQ_3:25;
          then
A152:     x1=x-1 by XREAL_1:233;
          1<=m by A149,FINSEQ_3:25;
          then 1+1<=m+x by A151,XREAL_1:7;
          then
A153:     2-1<=m+x-1 by XREAL_1:9;
          set mx=m+x1;
A154:     mx-'m=mx-m by NAT_1:11,XREAL_1:233;
          L+S-1<=L+S-0 by XREAL_1:10;
          then mx<=L+S by A152,A150,XXREAL_0:2;
          then
A155:     mx in Seg (S+L) by A152,A153;
          then r.mx=r/.mx by A30,PARTFUN1:def 6;
          then reconsider k=r.mx as Element of NAT;
A156:     B.mx in RNG by A59,A155,FUNCT_1:def 3;
A157:     Sum(LJ| (m-'1))< x1+1 by A9,A145,A152,MATRIXJ1:7;
          then m<=mx & Sum(LJ| (m-'1))<= mx-'m by A154,NAT_1:11,13;
          then
A158:     m<=k by A29,A149,A155;
A159:     m=k
          proof
            assume m<>k;
            then
A160:       m<k by A158,XXREAL_0:1;
            then reconsider k1=k-1 as Element of NAT by NAT_1:20;
A161:       k=k1+1;
            then m<=k1 by A160,NAT_1:13;
            then
A162:       Sum(LJ|m)<=Sum(LJ|k1) by POLYNOM3:18;
A163:       mx-'k<=mx-'m by A158,NAT_D:41;
            k-'1=k1 by A161,NAT_D:34;
            then Sum(LJ|k1) <=mx-'k by A29,A155;
            then Sum(LJ|m) <=mx-'k by A162,XXREAL_0:2;
            then Sum(LJ|m)<=x1 by A154,A163,XXREAL_0:2;
            hence thesis by A152,A157,A148,NAT_1:13;
          end;
A164:     mx-'m =Sum(LJ| (m-'1)) or mx-'m <>Sum(LJ| (m-'1));
          mx-'m<Sum(LJ|m) by A152,A154,A157,A148,NAT_1:13;
          then F.(B.mx)= b1.(mx-'m+1) by A57,A155,A159,A164;
          hence thesis by A146,A152,A154,A156,A144,FUNCT_1:def 6;
        end;
        F.(f/.n1) in im F & F|^1=F by RANKNULL:13,VECTSP11:19;
        then F.(f/.n1) in Lin(rngB1) by VECTSP_7:def 3;
        then consider L be Linear_Combination of rngB1 such that
A165:   F.(f/.n1) = Sum L by VECTSP_7:7;
        consider K be Linear_Combination of V1 such that
A166:   Carrier L=Carrier K and
A167:   Sum L=Sum K by VECTSP_9:8;
        Carrier L c= rngB1 by VECTSP_6:def 4;
        then consider M be Linear_Combination of RNG such that
A168:   F.(Sum M)=Sum K by A143,A166,VECTSP11:41,XBOOLE_1:1;
A169:   f.n1 in rng f by A141,FUNCT_1:def 3;
        then
A170:   fn in Rf by A142,XBOOLE_0:def 3;
A171:   not fn in RNG by A137,A142,A169,XBOOLE_0:3;
        not fn in RNG by A137,A142,A169,XBOOLE_0:3;
        then
A172:   RNG c= Rf\{fn} by XBOOLE_1:7,ZFMISC_1:34;
        Carrier M c= RNG & Carrier M=Carrier(-M) by VECTSP_6:38,def 4;
        then Carrier(-M) c= Rf\{fn} by A172;
        then reconsider M9=-M as Linear_Combination of Rf\{fn} by
VECTSP_6:def 4;
        set fnM=fn+Sum(M9);
A173:   fnM <> fn
        proof
          assume fnM=fn;
          then 0.V1 = fnM-fn by VECTSP_1:16
            .= Sum(M9)+(fn-fn) by RLVECT_1:def 3
            .= Sum(M9)+0.V1 by RLVECT_1:def 10
            .= Sum(M9) by RLVECT_1:def 4
            .= -Sum(M) by VECTSP_6:46;
          then 0.V1=Sum(M) by VECTSP_1:28;
          hence thesis by A140,A142,A165,A167,A168,RANKNULL:9;
        end;
        take ff=f+*(n1,fnM);
        set fnS=n1 .--> fnM;
A174:   Rf is linearly-independent by A138,VECTSP_7:def 3;
A175:   not fnM in Rf\{fn}
        proof
          card Rf <>0 by A169;
          then reconsider c1=card Rf-1 as Element of NAT by NAT_1:20;
          assume fnM in Rf\{fn};
          then
A176:     Rf\{fn}\/ {fnM} = Rf\{fn} by ZFMISC_1:40;
          c1+1=card Rf;
          then
A177:     card (Rf\{fn})= c1 by A170,STIRL2_1:55;
          card (Rf\{fn}\/ {fnM})=c1+1 by A174,A170,VECTSP11:40;
          hence thesis by A177,A176;
        end;
        not fnM in rng f
        proof
          assume fnM in rng f;
          then fnM in Rf by XBOOLE_0:def 3;
          hence thesis by A175,A173,ZFMISC_1:56;
        end;
        then
A178:   rng f misses {fnM} by ZFMISC_1:50;
        rng fnS={fnM} by FUNCOP_1:8;
        then f+*(fnS) is one-to-one by A135,A178,FUNCT_4:92;
        hence ff is one-to-one by A141,FUNCT_7:def 3;
A179:   dom ff=dom f by FUNCT_7:30;
        hence len ff=card C by A136,FINSEQ_3:29;
A180:   rng ff = (rng f)\{fn} \/{fnM} by A135,A141,A142,Lm1;
        thus RNG misses rng ff
        proof
          assume RNG meets rng ff;
          then consider x being object such that
A181:     x in RNG and
A182:     x in rng ff by XBOOLE_0:3;
          not x in (rng f)\{fn} by A137,A181,XBOOLE_0:3;
          then x in {fnM} by A180,A182,XBOOLE_0:def 3;
          then
A183:     x = fnM by TARSKI:def 1;
          not fnM in Rf by A175,A173,ZFMISC_1:56;
          hence thesis by A181,A183,XBOOLE_0:def 3;
        end;
A184:   Rf\{fn}\/{fnM} = (RNG\{fn})\/((rng f)\{fn})\/{fnM} by XBOOLE_1:42
          .= RNG\/((rng f)\{fn})\/{fnM} by A171,ZFMISC_1:57
          .= RNG\/ rng ff by A180,XBOOLE_1:4;
        then reconsider Rff=RNG\/ rng ff as finite Subset of V1;
        dim V1 = card Rf by A138,VECTSP_9:def 1
          .= card (RNG\/ rng ff) by A174,A170,A184,VECTSP11:40;
        then dim Lin(Rff)=dim V1 by A174,A170,A184,VECTSP11:40,VECTSP_9:26;
        then
A185:   (Omega).V1=(Omega).Lin(Rff) by VECTSP_9:28;
        Rf\{fn}\/{fnM} is linearly-independent by A174,A170,VECTSP11:40;
        hence RNG\/ rng ff is Basis of V1 by A184,A185,VECTSP_7:def 3;
        let i such that
A186:   i in dom ff and
A187:   i<= n1;
        per cases by A187,XXREAL_0:1;
        suppose
          i<n1;
          then ff.i=f.i & i<=n by FUNCT_7:32,NAT_1:13;
          hence thesis by A139,A179,A186;
        end;
        suppose
          i=n1;
          then ff.i=fnM by A179,A186,FUNCT_7:31;
          hence F.(ff.i) = F.(fn - Sum M) by VECTSP_6:46
            .= F.fn-F.(Sum M) by RANKNULL:8
            .= 0.V1 by A165,A167,A168,RLVECT_1:def 10;
        end;
      end;
    end;
A188: card (RNG\/C) = card RNG + card C - card (RNG/\C) by CARD_2:45
      .= card BAS-card (RNG/\C) by A128;
    then card (RNG\/C)+card (RNG/\C) =card BAS;
    then
A189: card (RNG\/C) <= card BAS by NAT_1:11;
A190: dim Lin BAS=card BAS by A4,VECTSP_9:26;
    then
A191: card (RNG\/C) >= card BAS by A130,A129,MATRLIN2:6;
    then
A192: card (RNG\/C) = card BAS by A189,XXREAL_0:1;
    dim V1= dim Lin(RNG\/C) by A130,A129,VECTSP_9:28;
    then
A193: RNG\/C is linearly-independent by A131,A190,A191,A189,MATRLIN2:5
,XXREAL_0:1;
A194: W[0]
    proof
      assume 0<= card C;
      card C=card Seg card C by FINSEQ_1:57;
      then Seg card C,C are_equipotent by CARD_1:5;
      then consider f be Function such that
A195: f is one-to-one and
A196: dom f = Seg card C and
A197: rng f = C by WELLORD2:def 4;
      reconsider f as FinSequence by A196,FINSEQ_1:def 2;
      reconsider f as FinSequence of V1 by A197,FINSEQ_1:def 4;
      take f;
      thus f is one-to-one & len f = card C by A195,A196,FINSEQ_1:def 3;
      RNG/\C={} by A188,A192;
      hence RNG misses rng f & RNG\/rng f is Basis of V1 by A129,A193,A197,
VECTSP_7:def 3,XBOOLE_0:def 7;
      let i;
      assume i in dom f & i<=0;
      hence thesis by FINSEQ_3:25;
    end;
    for n holds W[n] from NAT_1:sch 2(A194,A132);
    then consider f be FinSequence of V1 such that
A198: f is one-to-one and
A199: len f = card C and
A200: RNG misses rng f and
A201: RNG\/rng f is Basis of V1 and
A202: for i st i in dom f & i<= card C holds F.(f.i)=0.V1;
A203: rng (B^f)=rng B\/rng f by FINSEQ_1:31;
    now
      let x1,x2 be object such that
A204: x1 in dom B and
A205: x2 in dom B and
A206: B.x1 = B.x2;
      reconsider i1=x1,i2=x2 as Nat by A204,A205;
      r/.i1=r.i1 & r/.i2=r.i2 by A30,A59,A204,A205,PARTFUN1:def 6;
      then reconsider k1=r.i1,k2=r.i2 as Element of NAT;
A207: i1 -' k1 = Sum (LJ| (k1-'1)) implies F.(B.x1) = b1.(i1 -' k1+1)& i1
      -'k1+1 in dom b1 by A57,A59,A204;
A208: Q[i1,k1] by A29,A59,A204;
      then
A209: i1-'k1 =i1-k1 by XREAL_1:233;
A210: Q[i2,k2] by A29,A59,A205;
      then
A211: i2-'k2=i2-k2 by XREAL_1:233;
A212: k2-'1<=k2 by NAT_D:35;
A213: i1 -' k1 <> Sum (LJ| (k1-'1)) implies B.x1 = b1.(i1 -' k1) & i1-'k1
in dom b1 & min(LJ,i1-'k1)=k1 & (i1-'k1 < Sum (LJ|k1) implies F.(B.x1) = b1.(i1
-' k1+1) & i1 -' k1+1 in dom b1) & (i1-'k1 = Sum (LJ|k1) implies F.(B.x1) = 0.
      V1) by A57,A59,A204;
      k2 <= len LJ by A210,FINSEQ_3:25;
      then
A214: k2-'1 <=len LJ by A212,XXREAL_0:2;
      1<=k1 by A208,FINSEQ_3:25;
      then
A215: k1-'1=k1-1 by XREAL_1:233;
A216: i2 -' k2 <> Sum (LJ| (k2-'1)) implies B.x2 = b1.(i2 -' k2) & i2-'k2
in dom b1 & min(LJ,i2-'k2)=k2 & (i2-'k2 < Sum (LJ|k2) implies F.(B.x2) = b1.(i2
-' k2+1) & i2 -' k2+1 in dom b1) & (i2-'k2 = Sum (LJ|k2) implies F.(B.x2) = 0.
      V1) by A57,A59,A205;
      1<=k2 by A210,FINSEQ_3:25;
      then
A217: k2-'1=k2-1 by XREAL_1:233;
A218: i2 -' k2 = Sum (LJ| (k2-'1)) implies F.(B.x2) = b1.(i2 -' k2+1) &
      i2-'k2+1 in dom b1 by A57,A59,A205;
A219: k1-'1 <= k1 by NAT_D:35;
      k1<= len LJ by A208,FINSEQ_3:25;
      then
A220: k1-'1 <=len LJ by A219,XXREAL_0:2;
A221: dom LJ=dom J by MATRIXJ1:def 3;
      rng b1 is Basis of IM by MATRLIN:def 2;
      then
A222: rng b1 is linearly-independent Subset of IM by VECTSP_7:def 3;
A223: b1 is one-to-one by MATRLIN:def 2;
A224: i1-'k1 <= Sum (LJ|k1) & i2-'k2 <= Sum (LJ|k2) by A29,A59,A204,A205;
      now
        per cases by A224,XXREAL_0:1;
        suppose
A225:     i1 -' k1 = Sum (LJ| (k1-'1)) & i2 -' k2 = Sum (LJ| (k2-'1));
          then
A226:     F.(B.x2) = b1.(i2 -' k2+1) & i2-'k2+1 in dom b1 by A57,A59,A205;
          F.(B.x1) = b1.(i1 -' k1+1) & i1-'k1+1 in dom b1 by A57,A59,A204,A225;
          then
A227:     i1-'k1+1 = i2-'k2+1 by A206,A223,A226;
A228:     k2-'1 in dom LJ implies LJ.(k2-'1)<>0 by A7,A221;
          k1-'1 in dom LJ implies LJ.(k1-'1)<>0 by A7,A221;
          then k1-'1 = k2-'1 by A220,A214,A225,A227,A228,MATRIXJ1:11;
          hence i1=i2 by A215,A217,A209,A211,A227;
        end;
        suppose
A229:     i1 -' k1 = Sum (LJ| (k1-'1)) & i2 -' k2 <> Sum (LJ| (k2-'1))
          & i2-'k2 < Sum (LJ|k2);
          then
A230:     min(LJ,i2-'k2)=k2 by A57,A59,A205;
A231:     F.(B.x2) = b1.(i2-'k2+1) & i2-'k2+1 in dom b1 by A57,A59,A205,A229;
          F.(B.x1) = b1.(i1-'k1+1) & i1-'k1+1 in dom b1 by A57,A59,A204,A229;
          then
A232:     i1-'k1+1 = i2-'k2+1 by A206,A223,A231;
          k1-'1 <>0
          proof
            assume k1-'1=0;
            then LJ| (k1-'1)=<*>REAL;
            hence thesis by A29,A59,A205,A229,A232,RVSUM_1:72;
          end;
          then k1-'1 >=1 by NAT_1:14;
          then
A233:     k1-'1 in dom LJ by A220,FINSEQ_3:25;
          then LJ.(k1-'1)<>0 by A7,A221;
          hence i1=i2 by A229,A230,A232,A233,MATRIXJ1:6;
        end;
        suppose
          i1 -' k1 = Sum (LJ| (k1-'1)) & i2 -' k2 <> Sum (LJ| (k2-'1))
          & i2-'k2 = Sum (LJ|k2);
          then b1.(i1 -' k1+1) =0.IM & b1.(i1 -' k1+1) in rng b1 by A206,A207
,A216,FUNCT_1:def 3,VECTSP_4:11;
          hence i1 = i2 by A222,VECTSP_7:2;
        end;
        suppose
A234:     i2 -' k2 = Sum (LJ| (k2-'1)) & i1 -' k1 <> Sum (LJ| (k1-'1))
          & i1-'k1 < Sum (LJ|k1);
          then
A235:     min(LJ,i1-'k1)=k1 by A57,A59,A204;
A236:     F.(B.x1) = b1.(i1 -' k1+1) & i1-'k1+1 in dom b1 by A57,A59,A204,A234;
          F.(B.x2) = b1.(i2 -' k2+1) & i2-'k2+1 in dom b1 by A57,A59,A205,A234;
          then
A237:     i2-'k2+1 = i1-'k1+1 by A206,A223,A236;
          k2-'1 <>0
          proof
            assume k2-'1=0;
            then i1-'k1 =0 by A234,A237,RVSUM_1:72;
            hence thesis by A29,A59,A204,A234;
          end;
          then k2-'1 >=1 by NAT_1:14;
          then
A238:     k2-'1 in dom LJ by A214,FINSEQ_3:25;
          then LJ.(k2-'1)<>0 by A7,A221;
          hence i1=i2 by A234,A235,A237,A238,MATRIXJ1:6;
        end;
        suppose
          i2 -' k2 = Sum (LJ| (k2-'1)) & i1 -' k1 <> Sum (LJ| (k1-'1))
          & i1-'k1 = Sum (LJ|k1);
          then b1.(i2 -' k2+1) =0.IM & b1.(i2 -' k2+1) in rng b1 by A206,A213
,A218,FUNCT_1:def 3,VECTSP_4:11;
          hence i1= i2 by A222,VECTSP_7:2;
        end;
        suppose
A239:     i1 -' k1 <> Sum (LJ| (k1-'1)) & i2 -' k2 <> Sum (LJ| (k2-'1) );
          then i2-'k2 = i1-'k1 by A206,A213,A216,A223;
          then i2-k1=i1-k1 by A57,A59,A205,A213,A209,A211,A239;
          hence i1=i2;
        end;
      end;
      hence x1=x2;
    end;
    then B is one-to-one;
    then B^f is one-to-one by A198,A200,FINSEQ_3:91;
    then reconsider Bf=B^f as OrdBasis of V1 by A201,A203,MATRLIN:def 2;
    for i st i in dom Bf holds F.(Bf/.i) = 0.K * (Bf/.i) or i+1 in dom
    Bf & F.(Bf/.i) = 0.K * (Bf/.i) +Bf/.(i+1)
    proof
      let i such that
A240: i in dom Bf;
A241: Bf/.i=Bf.i by A240,PARTFUN1:def 6;
      per cases by A240,FINSEQ_1:25;
      suppose
A242:   i in dom B;
        then r/.i=r.i by A30,A59,PARTFUN1:def 6;
        then reconsider k=r.i as Element of NAT;
A243:   i-'k <= Sum (LJ|k) by A29,A59,A242;
A244:   Q[i,k] by A29,A59,A242;
        then 1<=k by FINSEQ_3:25;
        then k-'1=k-1 by XREAL_1:233;
        then k-'1+1=k;
        then LJ|k=(LJ| (k-'1))^<*LJ.k*> by A244,FINSEQ_5:10;
        then
A245:   dom LJ=dom J & Sum(LJ|k) = Sum (LJ| (k-'1))+LJ.k by MATRIXJ1:def 3
,RVSUM_1:74;
        per cases by A243,XXREAL_0:1;
        suppose
A246:     i -' k = Sum (LJ|k);
          then
A247:     i-'k <>Sum (LJ| (k-'1)) by A7,A244,A245;
          F.(Bf/.i) = F.(B.i) by A241,A242,FINSEQ_1:def 7
            .= 0.V1 by A57,A59,A242,A246,A247
            .= 0.K*(Bf/.i) by VECTSP_1:14;
          hence thesis;
        end;
        suppose
A248:     i -' k < Sum (LJ|k);
A249:     i -' k = Sum (LJ| (k-'1)) or i -' k <> Sum (LJ| (k-'1));
          then
A250:     F.(B.i)=b1.(i -' k+1) by A57,A59,A242,A248;
          dom J=dom LJ by MATRIXJ1:def 3;
          then
A251:     k <= L by A244,FINSEQ_3:25;
A252:     i-'k+1<=Sum(LJ|k) by A248,NAT_1:13;
A253:     i-'k=i-k by A244,XREAL_1:233;
A254:     i-'k+1 in dom b1 by A57,A59,A242,A248,A249;
          then
A255:     1<=i-'k+1 by FINSEQ_3:25;
          then
A256:     1+0<= i-k+1+k by A253,XREAL_1:7;
          i-'k+1<=S by A8,A254,FINSEQ_3:25;
          then i-k+1+k<=S+L by A251,A253,XREAL_1:7;
          then
A257:     i+1 in Seg(S+L) by A256;
          then r/.(i+1)=r.(i+1) by A30,PARTFUN1:def 6;
          then reconsider k1=r.(i+1) as Element of NAT;
          set i1=i+1;
A258:     dom B c= dom Bf by FINSEQ_1:26;
          1+k<=i-k+1+k by A255,A253,XREAL_1:7;
          then
A259:     k<=i+1 by NAT_1:13;
          then
A260:     i1-'k=i1-k by XREAL_1:233;
          Sum (LJ| (k-'1))<=i-'k+1 by A244,NAT_1:12;
          then
A261:     k<=k1 by A29,A244,A253,A257,A259,A260;
A262:     Q[i1,k1] by A29,A257;
A263:     k=k1
          proof
            assume
A264:       k<>k1;
            then
A265:       k<k1 by A261,XXREAL_0:1;
            then reconsider K1=k1-1 as Element of NAT by NAT_1:20;
A266:       i1-'k1 <= i1-'k by A261,NAT_D:41;
            i1-k1=i1-'k1 by A262,XREAL_1:233;
            then i1-'k1 <> i1-'k by A260,A264;
            then
A267:       i1-'k1 < i1-'k by A266,XXREAL_0:1;
A268:       k1=K1+1;
            then k<=K1 by A265,NAT_1:13;
            then
A269:       Sum(LJ|k)<=Sum(LJ|K1) by POLYNOM3:18;
            k1-'1=K1 by A268,NAT_D:34;
            then Sum(LJ|K1) <= i1-'k1 by A29,A257;
            then Sum(LJ|k)<= i1-'k1 by A269,XXREAL_0:2;
            hence thesis by A252,A253,A260,A267,XXREAL_0:2;
          end;
          Sum(LJ| (k-'1))<i-'k+1 by A244,NAT_1:13;
          then B.i1 = b1.(i -' k+1) by A57,A253,A257,A260,A263;
          then Bf.i1=b1.(i -' k+1) by A59,A257,FINSEQ_1:def 7;
          then Bf/.i1=b1.(i -' k+1) by A59,A257,A258,PARTFUN1:def 6;
          then F.(Bf/.i) = Bf/.i1 by A241,A242,A250,FINSEQ_1:def 7
            .= 0.V1+Bf/.i1 by RLVECT_1:def 4
            .= 0.K*(Bf/.i)+Bf/.i1 by VECTSP_1:14;
          hence thesis by A59,A257,A258;
        end;
      end;
      suppose
        ex j st j in dom f & i=len B + j;
        then consider j such that
A270:   j in dom f and
A271:   i=len B + j;
A272:   j<=len f by A270,FINSEQ_3:25;
        F.(Bf/.i) = F.(f.j) by A241,A270,A271,FINSEQ_1:def 7
          .= 0.V1 by A199,A202,A270,A272
          .= 0.K*(Bf/.i) by VECTSP_1:14;
        hence thesis;
      end;
    end;
    then consider
    J be non-empty FinSequence_of_Jordan_block of 0.K,K such that
A273: AutMt(F,Bf,Bf) = block_diagonal(J,0.K)by Th28;
    now
A274: dom (Len J)=dom J by MATRIXJ1:def 3;
      let i such that
A275: i in dom J;
      J.i<>{} by A275,FUNCT_1:def 9;
      hence (Len J).i <> 0 by A275,A274,MATRIXJ1:def 3;
    end;
    hence thesis by A273;
  end;
A276: P[0]
  proof
    reconsider J={} as FinSequence_of_Jordan_block of 0.K,K by Th10;
    let V1 be finite-dimensional VectSp of K;
    set b1 = the OrdBasis of V1;
    let F be nilpotent linear-transformation of V1,V1;
    assume deg F=0;
    then [#]V1 = {0.V1} by Th15
      .= the carrier of (0).V1 by VECTSP_4:def 3;
    then (0).V1 = (Omega).V1 by VECTSP_4:29;
    then
A277: 0 = dim V1 by VECTSP_9:29
      .= len b1 by MATRLIN2:21
      .= len AutMt(F,b1,b1) by MATRIX_0:def 2;
    take J,b1;
    thus AutMt(F,b1,b1) = {} by A277
      .= block_diagonal(J,0.K) by MATRIXJ1:22;
    thus thesis;
  end;
  for n holds P[n] from NAT_1:sch 2(A276,A1);
  then P[deg F];
  then consider
  J be FinSequence_of_Jordan_block of 0.K,K, b1 be OrdBasis of V1
  such that
A278: AutMt(F,b1,b1) = block_diagonal(J,0.K) and
A279: for i st i in dom J holds (Len J).i <> 0;
  now
    let x be object such that
A280: x in dom J;
    reconsider i=x as Element of NAT by A280;
    dom J=dom (Len J) & (Len J).i <> 0 by A279,A280,MATRIXJ1:def 3;
     then J.i is non empty by A280,MATRIXJ1:def 3;
    hence J.x is non empty;
  end;
  then J is non-empty;
  hence thesis by A278;
end;
