theorem
  for J be FinSequence_of_Jordan_block of 0.K,K for M be Matrix of len
  b1,len b1,K st M = block_diagonal(J,0.K) & len b1 > 0 for F be nilpotent
Function of V1,V1 st F = Mx2Tran(M,b1,b1) holds (ex i st i in dom J & len (J.i)
  = deg F) & for i st i in dom J holds len (J.i) <= deg F
proof
  let J be FinSequence_of_Jordan_block of 0.K,K;
  let M be Matrix of len b1,len b1,K such that
A1: M = block_diagonal(J,0.K) and
A2: len b1 > 0;
A3: len M=len b1 & len M=Sum Len J by A1,MATRIX_0:def 2;
  defpred P[Nat] means for i st i in dom J holds len (J.i) <= $1;
  set mm=min(Len J,len b1);
A4: dom J=dom (Len J) by MATRIXJ1:def 3;
  now
    let i such that
A5: i in dom J;
    len (J.i)=(Len J).i by A4,A5,MATRIXJ1:def 3;
    hence len (J.i) <= Sum Len J by A4,A5,POLYNOM3:4;
  end;
  then
A6: ex k st P[k];
  consider MIN be Nat such that
A7: P[MIN] and
A8: for m st P[m] holds MIN <= m from NAT_1:sch 5(A6);
  len b1 in Seg len b1 by A2,FINSEQ_1:3;
  then
A9: min(Len J,len b1) in dom Len J by A3,MATRIXJ1:def 1;
A10: ex i st i in dom J & len (J.i) = MIN
  proof
    assume
A11: for i st i in dom J holds len (J.i) <> MIN;
    len (J.mm)<=MIN by A9,A4,A7;
    then len (J.mm)<MIN by A9,A4,A11,XXREAL_0:1;
    then reconsider M1=MIN-1 as Element of NAT by NAT_1:20;
    now
      let i such that
A12:  i in dom J;
      len (J.i) <= MIN by A7,A12;
      then len (J.i) < M1+1 by A11,A12,XXREAL_0:1;
      hence len (J.i) <= M1 by NAT_1:13;
    end;
    then M1+1<=M1 by A8;
    hence thesis by NAT_1:13;
  end;
A13: (Len J) | (len Len J)=Len J by FINSEQ_1:58;
  let F be nilpotent Function of V1,V1 such that
A14: F = Mx2Tran(M,b1,b1);
  consider i such that
A15: i in dom J and
A16: len (J.i) = MIN by A10;
A17: (Len J).i=(Len J)/.i by A4,A15,PARTFUN1:def 6;
  set S=Sum ((Len J) | (i-'1));
  defpred P[Nat] means $1 in Seg MIN & $1 <> MIN implies (F|^$1).(b1/.(S+1))=
  b1/.(S+$1+1);
A18: len (J.i)=(Len J).i by A4,A15,MATRIXJ1:def 3;
  i<=len Len J by A4,A15,FINSEQ_3:25;
  then Sum ((Len J) |i) <= Sum ((Len J) | (len Len J)) by POLYNOM3:18;
  then
A19: dom b1=Seg len b1 & Seg Sum ((Len J) |i) c= Seg Sum Len J by A13,
FINSEQ_1:5,def 3;
  1<=i by A15,FINSEQ_3:25;
  then i-'1=i-1 by XREAL_1:233;
  then
A20: i=i-'1+1;
A21: for n st P[n] holds P[n+1]
  proof
    (Len J) |i=((Len J) | (i-'1))^<*MIN*> by A4,A15,A16,A18,A20,FINSEQ_5:10;
    then
A22: Sum ((Len J) |i)=S+MIN by RVSUM_1:74;
    let n such that
A23: P[n];
A24: (Len J) |i=Len J|i by MATRIXJ1:17;
    set n1=n+1;
    assume that
A25: n1 in Seg MIN and
A26: n1<>MIN;
A27: n1<=MIN by A25,FINSEQ_1:1;
    then n1<MIN by A26,XXREAL_0:1;
    then
A28: S+n1 < Sum ((Len J) |i) by A22,XREAL_1:6;
    S+n1 in Seg Sum ((Len J) |i) & min(Len J,S+n1) = i by A4,A15,A16,A18,A17
,A25,MATRIXJ1:10;
    then
A29: F.(b1/.(S+n1)) = 0.K*(b1/.(S+n1))+ (b1/.(S+n1+1)) by A1,A3,A14,A19,A28,A24
,Th24
      .= 0.V1+b1/.(S+n1+1) by VECTSP_1:14
      .= b1/.(S+n1+1) by RLVECT_1:def 4;
A30: n<MIN by A27,NAT_1:13;
    now
      per cases by NAT_1:14;
      suppose
        n=0;
        hence thesis by A29,VECTSP11:19;
      end;
      suppose
A31:    n>=1;
A32:    dom (F|^n)= the carrier of V1 by FUNCT_2:def 1;
        thus (F|^n1).(b1/.(S+1)) = ((F|^1)*(F|^n)).(b1/.(S+1)) by VECTSP11:20
          .= (F|^1).((F|^n).(b1/.(S+1))) by A32,FUNCT_1:13
          .= b1/.(S+n1+1) by A23,A30,A29,A31,VECTSP11:19;
      end;
    end;
    hence thesis;
  end;
A33: P[0];
A34: for n holds P[n] from NAT_1:sch 2(A33,A21);
A35: deg F >= MIN
  proof
    set D=deg F;
    rng b1 is Basis of V1 by MATRLIN:def 2;
    then
A36: rng b1 is linearly-independent Subset of V1 by VECTSP_7:def 3;
    assume
A37: D < MIN;
    then 1<=1+D & D+1<=MIN by NAT_1:11,13;
    then D+1 in Seg MIN;
    then S+(D+1) in Seg Sum ((Len J) |i) by A4,A15,A16,A18,A17,MATRIXJ1:10;
    then
A38: b1/.(S+D+1)=b1.(S+D+1) & b1.(S+D+1) in rng b1 by A3,A19,FUNCT_1:def 3
,PARTFUN1:def 6;
    D<>0
    proof
      assume D=0;
      then [#]V1={0.V1} by Th15;
      then (Omega).V1=(0).V1 by VECTSP_4:def 3;
      then dim V1=0 by VECTSP_9:29;
      hence thesis by A2,MATRLIN2:21;
    end;
    then D>=1 by NAT_1:14;
    then D in Seg MIN by A37;
    then b1/.(S+D+1) = (F|^D).(b1/.(S+1)) by A34,A37
      .= ZeroMap(V1,V1).(b1/.(S+1)) by Def5
      .= ((the carrier of V1)-->0.V1).(b1/.(S+1)) by GRCAT_1:def 7
      .= 0.V1;
    hence thesis by A38,A36,VECTSP_7:2;
  end;
  F|^MIN = ZeroMap(V1,V1) by A1,A14,A7,Th25;
  then deg F <= MIN by Def5;
  then deg F = MIN by A35,XXREAL_0:1;
  hence thesis by A7,A10;
end;
