
theorem Th29:
  for m being Nat st m > 0 for L being non empty ZeroStr, a being
AlgSequence of L, M being Matrix of m,1,L holds (for i being Nat st i < m holds
  M*(i+1,1) = a.i) implies mConv(a,m) = M
proof
  let m be Nat;
  assume
A1: m > 0;
  let L be non empty ZeroStr;
  let a be AlgSequence of L;
  let M be Matrix of m,1,L;
A2: width mConv(a, m) = 1 by A1,Th28
    .= width M by A1,MATRIX_0:23;
  assume
A3: for i being Nat st i < m holds M*(i+1,1) = a.i;
A4: for i,j being Nat st [i,j] in Indices mConv(a,m) holds (mConv(a,m))*(i,j
  ) = M*(i,j)
  proof
    let i,j be Nat;
    assume
A5: [i,j] in Indices mConv(a,m);
    then
A6: i in dom mConv(a, m) by ZFMISC_1:87;
    len mConv(a, m) = m by A1,Th28;
    then
A7: i in Seg m by A6,FINSEQ_1:def 3;
    then 0 < i by FINSEQ_1:1;
    then reconsider k=i-1 as Nat by NAT_1:20;
A8: j in Seg width mConv(a, m) by A5,ZFMISC_1:87;
    then
A9: 1 <= j by FINSEQ_1:1;
    j in Seg 1 by A1,A8,Th28;
    then
A10: j <= 1 by FINSEQ_1:1;
    k+1 <= m by A7,FINSEQ_1:1;
    then
A11: k < m by NAT_1:13;
    then M*(k+1,1) = a.k by A3
      .= (mConv(a, m))*(k+1,1) by A11,Th28
      .= (mConv(a, m))*(i,j) by A9,A10,XXREAL_0:1;
    hence thesis by A9,A10,XXREAL_0:1;
  end;
  len mConv(a,m) = m by A1,Th28
    .= len M by A1,MATRIX_0:23;
  hence thesis by A2,A4,MATRIX_0:21;
end;
