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

theorem Th1:
  diagonal_of_Matrix Jordan_block(L,n) = n |-> L
proof
  set B=Jordan_block(L,n);
A1: now
A2: [:Seg n,Seg n:]=Indices B by MATRIX_0:24;
    let i;
    assume 1<=i & i<=n;
    then
A3: i in Seg n;
    then
A4: [i,i] in [:Seg n,Seg n:] by ZFMISC_1:87;
    thus (diagonal_of_Matrix B).i = B*(i,i) by A3,MATRIX_3:def 10
      .= L by A4,A2,Def1
      .= (n|->L).i by A3,FINSEQ_2:57;
  end;
  len diagonal_of_Matrix B=n & len (n |-> L)=n by CARD_1:def 7,MATRIX_3:def 10;
  hence thesis by A1;
end;
