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 Th22:
  i in dom B1 & i <> len B1 implies Sum lmlt (Line(Jordan_block(L,
  len B1),i),B1)= L*(B1/.i)+B1/.(i+1)
proof
  assume that
A1: i in dom B1 and
A2: i<>len B1;
  set N=len B1;
A3: dom B1=Seg N by FINSEQ_1:def 3;
  i<=N by A1,FINSEQ_3:25;
  then i<N by A2,XXREAL_0:1;
  then 1<=i+1 & i+1<=N by NAT_1:11,13;
  then
A4: i+1 in dom B1 by A3;
  set ONE=1.(K,N);
A5: len Line(ONE,i+1)=width ONE by CARD_1:def 7;
  width ONE=N by MATRIX_0:24;
  then
A6: dom Line(ONE,i+1)= dom B1 by A5,FINSEQ_3:29;
  len (L*Line(ONE,i))=len Line(ONE,i) & len Line(ONE,i)=width ONE by
CARD_1:def 7,MATRIXR1:16;
  then dom (L*Line(ONE,i))=dom Line(ONE,i+1) by A5,FINSEQ_3:29;
  then
A7: dom lmlt(L*Line(ONE,i),B1) =dom B1 by A6,MATRLIN:12;
  dom lmlt(Line(ONE,i+1),B1) =dom B1 by A6,MATRLIN:12;
  then
A8: len lmlt (L*Line(ONE,i),B1)=len lmlt(Line(ONE,i+1),B1) by A7,FINSEQ_3:29;
  thus Sum lmlt (Line(Jordan_block(L,len B1),i),B1) = Sum lmlt (L*Line(ONE,i)+
  Line(ONE,i+1),B1) by A1,A2,A3,Th4
    .= Sum (lmlt (L*Line(ONE,i),B1)+lmlt(Line(ONE,i+1),B1)) by MATRLIN2:7
    .= (Sum lmlt (L*Line(ONE,i),B1))+(Sum lmlt(Line(ONE,i+1),B1)) by A8,
MATRLIN2:10
    .= (L*Sum lmlt (Line(ONE,i),B1))+(Sum lmlt(Line(ONE,i+1),B1)) by
MATRLIN2:13
    .= L*(B1/.i)+(Sum lmlt(Line(ONE,i+1),B1)) by A1,MATRLIN2:16
    .= L*(B1/.i)+B1/.(i+1) by A4,MATRLIN2:16;
end;
