
theorem Th36:
  for L being Field, m being Element of NAT st 0 < m for u,v,u1
being Matrix of m,L holds (for i,j being Nat st 1 <= i & i <= m & 1 <= j & j <=
  m holds (u * v)*(i,j) = emb(m,L) * (u1*(i,j))) implies u * v = emb(m,L) * u1
proof
  let L be Field;
  let m be Element of NAT;
  assume
A1: m > 0;
  let u,v,u1 be Matrix of m,L;
  assume
A2: for i,j being Nat st 1 <= i & i <= m & 1 <= j & j <= m holds (u * v)
  *(i,j) = emb(m,L) * (u1*(i,j));
A3: for i,j being Nat st [i,j] in Indices (u*v) holds (u*v)*(i,j) = (emb(m,L
  ) * u1)*(i,j)
  proof
    let i,j be Nat;
A4: [i,j] in Indices (u*v) implies 1 <= i & i <= m & 1 <= j & j <= m
    proof
      width u = m by MATRIX_0:24
        .= len v by MATRIX_0:24;
      then
A5:   width(u*v) = width(v) by MATRIX_3:def 4
        .= m by MATRIX_0:24;
      assume
A6:   [i,j] in Indices (u*v);
      then
A7:   j in Seg m by A5,ZFMISC_1:87;
      width u = m by MATRIX_0:24
        .= len v by MATRIX_0:24;
      then len(u*v) = len u by MATRIX_3:def 4
        .= m by MATRIX_0:24;
      then (u*v) is Matrix of m,m,L by A1,A5,MATRIX_0:20;
      then Indices (u*v) = [:Seg m,Seg m:] by A5,MATRIX_0:25;
      then i in Seg m by A6,ZFMISC_1:87;
      hence thesis by A7,FINSEQ_1:1;
    end;
    assume
A8: [i,j] in Indices (u*v);
    then i in Seg m & j in Seg m by A4;
    then [i,j] in [:Seg m, Seg m:] by ZFMISC_1:87;
    then [i,j] in Indices u1 by MATRIX_0:24;
    then (emb(m,L) * u1)*(i,j) = emb(m,L) * (u1*(i,j)) by MATRIX_3:def 5;
    hence thesis by A2,A8,A4;
  end;
A9: width(emb(m,L) * u1) = width(u1) by MATRIX_3:def 5
    .= m by MATRIX_0:24;
  width u = m by MATRIX_0:24
    .= len v by MATRIX_0:24;
  then
A10: width(u*v) = width(v) by MATRIX_3:def 4
    .= m by MATRIX_0:24;
  width u = m by MATRIX_0:24
    .= len v by MATRIX_0:24;
  then
A11: len(u*v) = len u by MATRIX_3:def 4
    .= m by MATRIX_0:24;
  len (emb(m,L) * u1) = len u1 by MATRIX_3:def 5
    .= m by MATRIX_0:24;
  hence thesis by A11,A9,A10,A3,MATRIX_0:21;
end;
