reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;

theorem Th3:
  Jordan_block(L,n) is invertible iff n = 0 or L <> 0.K
proof
  set B=Jordan_block(L,n);
A1: B is invertible implies L <> 0.K or n=0
  proof
    assume B is invertible;
    then
A2: Det B<>0.K by LAPLACE:34;
    assume
A3: L=0.K;
    assume n<>0;
    then
A4: n in Seg n by FINSEQ_1:3;
    then dom (n|->L)=Seg n & (n|->L).n=L by FINSEQ_2:57,124;
    then 0.K = Product (n|->L) by A3,A4,FVSUM_1:82
      .= (power K).(L,n) by MATRIXJ1:5;
    hence thesis by A2,Th2;
  end;
  n = 0 or L <> 0.K implies B is invertible
  proof
    assume
A5: n=0 or L<>0.K;
    assume not B is invertible;
    then 0.K = Det B by LAPLACE:34
      .= (power K).(L,n) by Th2
      .= Product (n|->L) by MATRIXJ1:5;
    then
A6: ex k be Nat st k in dom (n|->L) & (n|->L).k=0. K by FVSUM_1:82;
    dom (n|->L)=Seg n by FINSEQ_2:124;
    hence thesis by A5,A6,FINSEQ_2:57;
  end;
  hence thesis by A1;
end;
