
theorem Th35:
  for L being Field, m,n being Nat st m > 0 for M being Matrix of
  m,n,L holds 1.(L,m) * M = M
proof
  let L be Field, m,n be Nat;
  assume
A1: m > 0;
  let M be Matrix of m,n,L;
A2: width 1.(L,m) = m by A1,MATRIX_0:23
    .= len M by A1,MATRIX_0:23;
  set M2 = 1.(L,m) * M;
A3: len M = m by A1,MATRIX_0:23;
  len 1.(L,m) = m by A1,MATRIX_0:23;
  then
A4: m = len M2 by A2,MATRIX_3:def 4;
A5: now
    let i,j be Nat;
    assume
A6: [i,j] in Indices M;
    then
A7: i in dom M by ZFMISC_1:87;
    dom M = Seg len M by FINSEQ_1:def 3
      .= dom M2 by A3,A4,FINSEQ_1:def 3;
    then Indices M = Indices M2 by A2,MATRIX_3:def 4;
    then
A8: M2*(i,j) = Line(1.(L,m),i) "*" Col(M,j) by A2,A6,MATRIX_3:def 4
      .= Sum(mlt(Line(1.(L,m),i), Col(M,j))) by FVSUM_1:def 9;
    len Line(1.(L,m),i) = width 1.(L,m) by MATRIX_0:def 7
      .= m by MATRIX_0:24;
    then
A9: dom Line(1.(L,m),i) = Seg(m) by FINSEQ_1:def 3;
A10: len M = m by A1,MATRIX_0:23;
    then
A11: i in dom Line(1.(L,m),i) by A7,A9,FINSEQ_1:def 3;
A12: Indices 1.(L,m) = [:Seg m,Seg m:] by A1,MATRIX_0:23;
    then
A13: [i,i] in Indices 1.(L,m) by A9,A11,ZFMISC_1:87;
A14: for k being Nat st k in dom Line(1.(L,m),i) & k<>i holds Line(1.(L,m)
    ,i).k = 0.L
    proof
      let k be Nat;
      assume that
A15:  k in dom Line(1.(L,m),i) and
A16:  k<>i;
A17:  [i,k] in Indices 1.(L,m) by A9,A11,A12,A15,ZFMISC_1:87;
      k in Seg width 1.(L,m) by A9,A15,MATRIX_0:24;
      then Line(1.(L,m),i).k = 1.(L,m)*(i,k) by MATRIX_0:def 7
        .= 0.L by A16,A17,MATRIX_1:def 3;
      hence thesis;
    end;
    len Col(M,j) = len M by MATRIX_0:def 8
      .= m by A1,MATRIX_0:23;
    then dom Col(M,j) = Seg(m) by FINSEQ_1:def 3;
    then
A18: i in dom Col(M,j) by A7,A10,FINSEQ_1:def 3;
    i in Seg width 1.(L,m) by A9,A11,MATRIX_0:24;
    then
A19: Line(1.(L,m),i).i = 1.(L,m)*(i,i) by MATRIX_0:def 7
      .= 1.L by A13,MATRIX_1:def 3;
    i in dom Line(1.(L,m),i) by A7,A10,A9,FINSEQ_1:def 3;
    then Sum(mlt(Line(1.(L,m),i),Col(M,j))) = (Col(M,j)).i by A19,A14,A18,
MATRIX_3:17;
    hence M*(i,j) = M2*(i,j) by A8,A7,MATRIX_0:def 8;
  end;
  width M = width M2 by A2,MATRIX_3:def 4;
  hence thesis by A3,A4,A5,MATRIX_0:21;
end;
