
theorem Th39:
  for L being Field, m being Element of NAT st m > 0 for x being
Element of L st x is_primitive_root_of_degree m holds VM(x,m) * VM(x",m) = emb(
  m,L) * 1.(L,m)
proof
  let L be Field, m be Element of NAT;
  assume
A1: m > 0;
  let x be Element of L;
  assume
A2: x is_primitive_root_of_degree m;
  for i,j being Nat st i >= 1 & i <= m & j >= 1 & j <= m holds (VM(x,m) *
  VM(x",m))*(i,j) = emb(m,L) * (1.(L,m)*(i,j))
  proof
    let i,j be Nat;
A3: i in NAT & j in NAT by ORDINAL1:def 12;
    ex G being FinSequence of L st (dom G = Seg m & for k being Nat st k
    in Seg m holds G.k = pow(x, (i-j)*(k-1)))
    proof
      defpred P[Nat,set] means $2 = pow(x, (i-j)*($1-1));
A4:   for n being Nat st n in Seg m holds ex x being Element of L st P[n,x ];
      ex G be FinSequence of L st dom G = Seg m & for nn be Nat st nn in
      Seg m holds P[nn,G.nn] from FINSEQ_1:sch 5(A4);
      hence thesis;
    end;
    then consider s being FinSequence of L such that
A5: dom s = Seg m and
A6: for k being Nat st k in Seg m holds s.k = pow(x, (i-j)*(k-1));
A7: len s = m by A5,FINSEQ_1:def 3;
A8: for k being Nat st 1 <= k & k <= m holds s/.k = pow(x, (i-j)*(k-1))
    proof
      let k be Nat;
      assume
A9:   1 <= k & k <= m;
      then
A10:  k in dom s by A5;
      k in Seg m by A9;
      then pow(x, (i-j)*(k-1)) = s.k by A6
        .= s/.k by A10,PARTFUN1:def 6;
      hence thesis;
    end;
A11: Indices 1.(L,m) = [:Seg m, Seg m:] by MATRIX_0:24;
    assume that
A12: 1 <= i & i <= m and
A13: 1 <= j & j <= m;
    per cases;
    suppose
A14:  i = j;
A15:  for k being Element of NAT st 1 <= k & k <= m holds s/.k = 1.L
      proof
        let k be Element of NAT;
        assume 1 <= k & k <= m;
        then s/.k = pow(x, (i-j)*(k-1)) by A8
          .= 1.L by A14,Th13;
        hence thesis;
      end;
      i in Seg m by A12;
      then
A16:  [i,i] in Indices 1.(L,m) by A11,ZFMISC_1:87;
      (VM(x,m) * VM(x",m))*(i,j) = Sum s by A2,A3,A12,A13,A7,A8,Th37
        .= m * 1.L by A7,A15,Th4
        .= emb(m,L) * 1.L;
      hence thesis by A14,A16,MATRIX_1:def 3;
    end;
    suppose
A17:  i <> j;
      i in Seg m & j in Seg m by A12,A13;
      then
A18:  [i,j] in Indices 1.(L,m) by A11,ZFMISC_1:87;
      (VM(x,m) * VM(x",m))*(i,j) = 0.L by A2,A3,A12,A13,A17,Th38
        .= emb(m,L) * 0.L;
      hence thesis by A17,A18,MATRIX_1:def 3;
    end;
  end;
  hence thesis by A1,Th36;
end;
