reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;
reserve V1,V2 for finite-dimensional VectSp of K,
  W1,W2 for Subspace of V1,
  U1 ,U2 for Subspace of V2,
  b1 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,
  bw1 for OrdBasis of W1,
  bw2 for OrdBasis of W2,
  Bu1 for FinSequence of U1,
  Bu2 for FinSequence of U2;

theorem Th23:
  for M be Matrix of len b1,len B2,K st M = Jordan_block(L,n) for
i st i in dom b1 holds (i = len b1 implies Mx2Tran(M,b1,B2).(b1/.i) = L*(B2/.i)
  ) & (i <> len b1 implies Mx2Tran(M,b1,B2).(b1/.i) = L*(B2/.i)+ B2/.(i+1))
proof
  set ONE=1.(K,len b1);
  set J=Jordan_block(L,n);
  let M be Matrix of len b1,len B2,K such that
A1: M = Jordan_block(L,n);
A2: len M=n by A1,MATRIX_0:24;
  len ONE=len b1 by MATRIX_0:def 2;
  then
A3: dom ONE=dom b1 by FINSEQ_3:29;
  let i such that
A4: i in dom b1;
A5: len M=len b1 by MATRIX_0:25;
A6: Mx2Tran(M,b1,B2).(b1/.i) = Sum lmlt (Line(LineVec2Mx(b1/.i|--b1) * M,1)
  ,B2) by MATRLIN2:def 3
    .= Sum lmlt (Line(LineVec2Mx(Line(ONE,i))*M,1),B2) by A4,MATRLIN2:19
    .= Sum lmlt (Line(LineVec2Mx(Line(ONE*M,i)),1),B2) by A4,A3,A5,MATRIX_0:24
,MATRLIN2:35
    .= Sum lmlt (Line(LineVec2Mx(Line(M,i)),1),B2) by A5,MATRIXR2:68
    .= Sum lmlt (Line(M,i),B2) by MATRIX15:25;
  dom b1=Seg len b1 by FINSEQ_1:def 3;
  then n<>0 by A4,A5,A2;
  then
A7: width J=n & width J=len B2 by A1,A5,A2,MATRIX_0:20;
  then dom B2=dom b1 by A5,A2,FINSEQ_3:29;
  hence thesis by A1,A4,A5,A2,A7,A6,Th21,Th22;
end;
