theorem
  for J be FinSequence_of_Jordan_block of L,K for M be Matrix of len b1,
  len b1,K st M = block_diagonal(J,0.K) holds Mx2Tran(M,b1,b1) is nilpotent iff
  len b1 = 0 or L = 0.K
proof
  let J be FinSequence_of_Jordan_block of L,K;
  let M be Matrix of len b1,len b1,K such that
A1: M = block_diagonal(J,0.K);
  set MT=Mx2Tran(M,b1,b1);
  thus MT is nilpotent implies len b1 = 0 or L = 0.K
  proof
    set B=len b1;
    set m=min(Len J,B);
    set S=Sum (Len J|m);
    assume MT is nilpotent;
    then reconsider MT as nilpotent linear-transformation of V1,V1;
    rng b1 is Basis of V1 by MATRLIN:def 2;
    then
A2: rng b1 is linearly-independent Subset of V1 by VECTSP_7:def 3;
    assume
A3: B<>0;
    then S in dom b1 by A1,Lm3;
    then b1.S in rng b1 & b1/.S=b1.S by FUNCT_1:def 3,PARTFUN1:def 6;
    then
A4: b1/.S <>0.V1 by A2,VECTSP_7:2;
    ((power K).(L,deg MT))*(b1/.S) = (MT|^(deg MT)).(b1/.S) by A1,A3,Lm3
      .= ZeroMap(V1,V1).(b1/.S) by Def5
      .= ((the carrier of V1)-->0.V1).(b1/.S) by GRCAT_1:def 7
      .= 0.V1
      .= 0.K * (b1/.S) by VECTSP_1:14;
    then 0.K = (power K).(L,deg MT) by A4,VECTSP10:4
      .= Product (deg MT|->L) by MATRIXJ1:5;
    then
A5: ex k be Nat st k in dom (deg MT|->L) & (deg MT|->L).k=0.K
    by FVSUM_1:82;
    dom (deg MT|->L)=Seg deg MT by FINSEQ_2:124;
    hence thesis by A5,FINSEQ_2:57;
  end;
  assume
A6: len b1=0 or L=0.K;
  per cases by A6;
  suppose
    len b1=0;
    then dim V1=0 by MATRLIN2:21;
    then (Omega).V1=(0).V1 by VECTSP_9:29;
    then
A7: the carrier of V1 = {0.V1} by VECTSP_4:def 3;
    rng (MT|^1) c= the carrier of V1 by RELAT_1:def 19;
    then rng (MT|^1)={0.V1} by A7,ZFMISC_1:33;
    then (MT|^1) = (dom (MT|^1))-->0.V1 by FUNCOP_1:9
      .= (the carrier of V1)-->0.V1 by FUNCT_2:def 1
      .= ZeroMap(V1,V1) by GRCAT_1:def 7;
    hence thesis by Th13;
  end;
  suppose
A8: L=0.K;
    now
A9:   dom J=dom (Len J) by MATRIXJ1:def 3;
      let i such that
A10:  i in dom J;
      len (J.i)=(Len J).i by A10,A9,MATRIXJ1:def 3;
      hence len (J.i) <= Sum Len J by A10,A9,POLYNOM3:4;
    end;
    then MT|^(Sum Len J)=ZeroMap(V1,V1) by A1,A8,Th25;
    hence thesis by Th13;
  end;
end;
