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

theorem Th6:
  for F be Element of n-tuples_on the carrier of K st i in Seg n
holds ( i <> n implies Line(Jordan_block(L,n),i) "*" F = L * (F/.i)+ F/.(i+1))&
  ( i = n implies Line(Jordan_block(L,n),i) "*" F = L * (F/.i))
proof
  let F be Element of n-tuples_on the carrier of K such that
A1: i in Seg n;
  set J=Jordan_block(L,n);
A2: width J=n by MATRIX_0:24;
  then
A3: Line(J,i).i=J*(i,i) by A1,MATRIX_0:def 7;
A4: Indices J=[:Seg n,Seg n:] by MATRIX_0:24;
  then
A5: [i,i] in Indices J by A1,ZFMISC_1:87;
  reconsider N=n as Element of NAT by ORDINAL1:def 12;
  set Li=Line(J,i);
  reconsider Li,f=F as Element of N-tuples_on the carrier of K by MATRIX_0:24;
A6: dom f=Seg n by FINSEQ_2:124;
  then
A7: f.i=f/.i by A1,PARTFUN1:def 6;
A8: dom mlt(Li,f)=Seg n by FINSEQ_2:124;
  thus i <> n implies Line(J,i) "*" F = L * (F/.i)+ F/.(i+1)
  proof
A9: now
      let j such that
A10:  j in Seg n and
A11:  j<>i & j<>i+1;
      [i,j] in Indices J by A1,A4,A10,ZFMISC_1:87;
      then
A12:  0.K = J*(i,j) by A11,Def1
        .= Li.j by A2,A10,MATRIX_0:def 7;
      f.j=f/.j by A6,A10,PARTFUN1:def 6;
      hence mlt(Li,f).j = 0.K * (f/.j) by A10,A12,FVSUM_1:61
        .= 0.K;
    end;
A13: [i,i] in Indices J by A1,A4,ZFMISC_1:87;
    assume
A14: i<>n;
    i<=n by A1,FINSEQ_1:1;
    then i<n by A14,XXREAL_0:1;
    then 1<=i+1 & i+1<=n by NAT_1:11,13;
    then
A15: i+1 in Seg n;
    then
A16: [i,i+1] in Indices J by A1,A4,ZFMISC_1:87;
A17: f.i=f/.i & J*(i,i)=Li.i by A1,A2,A6,MATRIX_0:def 7,PARTFUN1:def 6;
A18: mlt(Li,f)/.i = mlt(Li,f).i by A1,A8,PARTFUN1:def 6
      .= J*(i,i)*f/.i by A1,A17,FVSUM_1:61
      .= L*(f/.i) by A13,Def1;
A19: i+1>i by NAT_1:13;
A20: f.(i+1)=f/.(i+1) & J*(i,i+1)=Li.(i+1) by A2,A6,A15,MATRIX_0:def 7
,PARTFUN1:def 6;
    mlt(Li,f)/.(i+1) = mlt(Li,f).(i+1) by A8,A15,PARTFUN1:def 6
      .= J*(i,i+1)*f/.(i+1) by A15,A20,FVSUM_1:61
      .= 1_K*(f/.(i+1)) by A16,Def1
      .= f/.(i+1);
    hence thesis by A1,A8,A15,A19,A18,A9,MATRIX15:7;
  end;
  assume
A21: i=n;
  now
    let j such that
A22: j in Seg n and
A23: j<>i;
A24: f.j=f/.j by A6,A22,PARTFUN1:def 6;
    j<=n by A22,FINSEQ_1:1;
    then
A25: j<i+1 by A21,NAT_1:13;
A26: [i,j] in Indices J by A1,A4,A22,ZFMISC_1:87;
    Line(J,i).j = J*(i,j) by A2,A22,MATRIX_0:def 7
      .= 0.K by A23,A25,A26,Def1;
    hence mlt(Line(J,i),f).j = 0.K*f/.j by A8,A22,A24,FVSUM_1:61
      .= 0.K;
  end;
  hence Line(J,i)"*"F = mlt(Line(J,i),f).i by A1,A8,MATRIX_3:12
    .= J*(i,i)*(f/.i) by A1,A8,A3,A7,FVSUM_1:61
    .= L*(F/.i) by A5,Def1;
end;
