theorem
  i in dom R implies Sum lmlt(Line(1.(K,len R),i),R) = R/.i
proof
  set ONE=1.(K,len R);
  set L=Line(ONE,i);
  set M=lmlt(L,R);
A1: width ONE=len R by MATRIX_0:24;
  len L=width ONE by CARD_1:def 7;
  then dom L=dom R by A1,FINSEQ_3:29;
  then
A2: dom M=dom R by MATRLIN:12;
  then
A3: len M=len R by FINSEQ_3:29;
  consider f be sequence of  the carrier of V1 such that
A4: Sum M= f.(len M) and
A5: f.0 = 0.V1 and
A6: for j be Nat, v1 st j < len M & v1 = M.(j + 1) holds f.(
  j + 1) = f.j + v1 by RLVECT_1:def 12;
  defpred Q[Nat] means $1<=len M implies f.$1=R/.i;
  defpred P[Nat] means $1<i implies f.$1=0.V1;
  assume
A7: i in dom R;
  then
A8: 1<=i by FINSEQ_3:25;
  len ONE=len R by MATRIX_0:24;
  then
A9: dom R=dom ONE by FINSEQ_3:29;
A10: for n st i<=n holds Q[n] implies Q[n+1]
  proof
    let n such that
A11: i <=n;
    set n1=n+1;
A12: i<n1 by A11,NAT_1:13;
    reconsider N=n as Element of NAT by ORDINAL1:def 12;
    assume
A13: Q[n];
    assume
A14: n1<=len M;
    then
A15: n<len M by NAT_1:13;
A16: 1<=n1 by NAT_1:11;
    then n1 in Seg len R by A3,A14;
    then L.n1=ONE*(i,n1) & [i,n1] in Indices ONE by A7,A1,A9,MATRIX_0:def 7
,ZFMISC_1:87;
    then
A17: L.n1=0.K by A12,MATRIX_1:def 3;
A18: n1 in dom R by A2,A14,A16,FINSEQ_3:25;
    then R.n1=R/.n1 by PARTFUN1:def 6;
    then M.n1 = 0.K * R/.n1 by A2,A18,A17,FUNCOP_1:22
      .= 0.V1 by VECTSP_1:14;
    hence f.n1 = f.N + 0.V1 by A6,A15
      .= R/.i by A13,A14,NAT_1:13,RLVECT_1:def 4;
  end;
A19: i<=len M by A7,A2,FINSEQ_3:25;
A20: for n st P[n] holds P[n+1]
  proof
    let n such that
A21: P[n];
    reconsider N=n as Element of NAT by ORDINAL1:def 12;
    set n1=n+1;
    assume
A22: n1<i;
    then n1<len M by A19,XXREAL_0:2;
    then
A23: n<len M by NAT_1:13;
A24: 1<=n1 & n1<=len R by A19,A3,A22,NAT_1:11,XXREAL_0:2;
    then n1 in Seg len R;
    then L.n1=ONE*(i,n1) & [i,n1] in Indices ONE by A7,A1,A9,MATRIX_0:def 7
,ZFMISC_1:87;
    then
A25: L.n1=0.K by A22,MATRIX_1:def 3;
A26: n1 in dom R by A24,FINSEQ_3:25;
    then R.n1=R/.n1 by PARTFUN1:def 6;
    then M.n1 = 0.K * R/.n1 by A2,A26,A25,FUNCOP_1:22
      .= 0.V1 by VECTSP_1:14;
    hence f.n1 = f.N + 0.V1 by A6,A23
      .= 0.V1 by A21,A22,NAT_1:13,RLVECT_1:def 4;
  end;
A27: P[0] by A5;
A28: for n holds P[n] from NAT_1:sch 2(A27,A20);
A29: Q[i]
  proof
    i in Seg len R by A8,A19,A3;
    then L.i=ONE*(i,i) & [i,i] in Indices ONE by A7,A1,A9,MATRIX_0:def 7
,ZFMISC_1:87;
    then
A30: L.i=1_K by MATRIX_1:def 3;
    reconsider i1=i-1 as Element of NAT by A8,NAT_1:21;
A31: i1+1=i;
    then i1<i by NAT_1:13;
    then
A32: f.i1=0.V1 by A28;
    assume i<=len M;
    then
A33: i1<len M by A31,NAT_1:13;
    R.i=R/.i by A7,PARTFUN1:def 6;
    then M.i = 1_K * R/.i by A7,A2,A30,FUNCOP_1:22
      .= R/.i;
    then f.(i1+1) = f.i1+R/.i by A6,A33;
    hence thesis by A32,RLVECT_1:def 4;
  end;
  for n st i<=n holds Q[n] from NAT_1:sch 8(A29,A10);
  hence thesis by A19,A4;
end;
