theorem Th24:
  for J be FinSequence_of_Jordan_block of L,K for M be Matrix of
len b1,len B2,K st M = block_diagonal(J,0.K) for i,m st i in dom b1 & m = min(
Len J,i) holds (i = Sum (Len J|m) implies Mx2Tran(M,b1,B2).(b1/.i) = L*(B2/.i))
& (i <> Sum (Len J|m) implies Mx2Tran(M,b1,B2).(b1/.i) = L*(B2/.i)+ B2/.(i+1))
proof
  let J be FinSequence_of_Jordan_block of L,K;
  let M be Matrix of len b1,len B2,K such that
A1: M = block_diagonal(J,0.K);
A2: dom b1=Seg len b1 & len M=Sum Len J by A1,FINSEQ_1:def 3,MATRIXJ1:def 5;
  let i,m such that
A3: i in dom b1 and
A4: m = min(Len J,i);
  set Sm=Sum (Len J|m);
A5: 1<=i by A3,FINSEQ_3:25;
A6: i<=len b1 by A3,FINSEQ_3:25;
  then
A7: len M=len b1 by A5,MATRIX_0:23;
  then
A8: m in dom Len J by A3,A4,A2,MATRIXJ1:def 1;
  then
A9: (Len J).m = len (J.m) by MATRIXJ1:def 3;
  set S=Sum (Len J| (m-'1));
  set iS=i-'S;
  set BBB=(B2|Sum(Width J|m))/^Sum (Width J| (m-'1));
A10: Mx2Tran(M,b1,B2).(b1/.i) = Sum lmlt(Line(J.m,iS),BBB) by A1,A3,A4,Th20;
A11: width M=Sum Width J by A1,MATRIXJ1:def 5;
A12: len BBB = width (J.m) by A1,A3,A4,Th20;
A13: (Len J) |m = Len (J|m) by MATRIXJ1:17;
  then
A14: i<=Sm by A3,A4,A7,A2,MATRIXJ1:def 1;
A15: (Len J) | (m-'1) = Len (J| (m-'1)) by MATRIXJ1:17;
  then S<i by A3,A4,A7,A2,MATRIXJ1:7;
  then
A16: iS=i-S by XREAL_1:233;
A17: m-'1=m-1 by A3,A4,A7,A2,MATRIXJ1:7;
  then (Len J) | (m-'1+1)=(Len J| (m-'1))^<*(Len J).m*> by A15,A8,FINSEQ_5:10;
  then
A18: Sm=S+len (J.m) by A13,A17,A9,RVSUM_1:74;
  then S+iS<=S+len (J.m) by A3,A4,A13,A7,A2,A16,MATRIXJ1:def 1;
  then
A19: iS<=len (J.m) by XREAL_1:6;
  dom Len J=dom J by MATRIXJ1:def 3;
  then consider n such that
A20: J.m = Jordan_block(L,n) by A8,Def3;
  m in NAT & m <= len Len J by A8,FINSEQ_3:25;
  then Sm <= Sum ((Len J) | (len Len J)) by A13,POLYNOM3:18;
  then
A21: Sm <= Sum Len J by FINSEQ_1:58;
A22: Width J=Len J by MATRIXJ1:46;
  then
A23: (Len J).m=width (J.m) by A8,MATRIXJ1:def 4;
  width M=len B2 by A5,A6,MATRIX_0:23;
  then
A24: len (B2|Sm) =Sm by A22,A11,A21,FINSEQ_1:59;
  then
A25: i in dom (B2|Sm) by A5,A14,FINSEQ_3:25;
A26: len (J.m)=n by A20,MATRIX_0:24;
  iS<>0 by A3,A4,A15,A7,A2,A16,MATRIXJ1:7;
  then 1<=iS by NAT_1:14;
  then
A27: iS in dom BBB by A12,A9,A23,A19,FINSEQ_3:25;
  then
A28: BBB/.iS = (B2|Sum(Width J|m))/.(Sum (Width J| (m-'1))+iS) by FINSEQ_5:27
    .= (B2|Sum(Width J|m))/.(S+iS) by MATRIXJ1:46
    .= (B2|Sum((Len J) |m))/.i by A22,A16,MATRIXJ1:21
    .= B2/.i by A13,A25,FINSEQ_4:70;
  hence i = Sm implies Mx2Tran(M,b1,B2).(b1/.i) = L*(B2/.i) by A10,A12,A9,A16
,A23,A18,A27,A20,A26,Th21;
  assume
A29: i<>Sm;
  then i<Sm by A14,XXREAL_0:1;
  then 1<=i+1 & i+1<=Sm by NAT_1:11,13;
  then
A30: i+1 in dom (B2|Sm) by A24,FINSEQ_3:25;
A31: iS<>len (J.m) by A16,A18,A29;
  then iS<len (J.m) by A19,XXREAL_0:1;
  then 1<=iS+1 & iS+1<=len (J.m) by NAT_1:11,13;
  then iS+1 in dom BBB by A12,A9,A23,FINSEQ_3:25;
  then BBB/.(iS+1) = (B2|Sum(Width J|m))/.(Sum (Width J| (m-'1))+(iS+1)) by
FINSEQ_5:27
    .= (B2|Sum(Width J|m))/.(Sum ((Len J) | (m-'1))+(iS+1)) by A22,MATRIXJ1:21
    .= (B2|Sum((Len J) |m))/.(i+1) by A15,A22,A16,MATRIXJ1:21
    .= B2/.(i+1) by A13,A30,FINSEQ_4:70;
  hence thesis by A10,A12,A9,A23,A27,A20,A26,A28,A31,Th22;
end;
