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;
