
theorem Th38:
  for L being Field, m,i,j being Element of NAT, x being Element
  of L st i <> j & 1 <= i & i <= m & 1 <= j & j <= m & x
  is_primitive_root_of_degree m holds (VM(x,m) * VM(x",m))*(i,j) = 0.L
proof
  let L be Field, m,i,j be Element of NAT, x be Element of L;
  assume that
A1: i <> j and
A2: 1 <= i & i <= m & 1 <= j & j <= m and
A3: x is_primitive_root_of_degree m;
A4: x <> 0.L by A3,Th30;
  then
A5: pow(x, m*(i-j)) = pow(pow(x, m), i-j) by Th26
    .= pow(x|^m, i-j) by Def2
    .= pow(1.L, i-j) by A3
    .= 1.L by Th16;
  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));
A6: 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(A6);
    hence thesis;
  end;
  then consider s being FinSequence of L such that
A7: dom s = Seg m and
A8: for k being Nat st k in Seg m holds s.k = pow(x,(i-j)*(k-1));
A9: for k being Nat st 1 <= k & k <= m holds s/.k = pow(x,(i-j)*(k-1))
  proof
    let k be Nat;
    assume
A10: 1 <= k & k <= m;
    then
A11: k in dom s by A7;
    k in Seg m by A10;
    then pow(x, (i-j)*(k-1)) = s.k by A8
      .= s/.k by A11,PARTFUN1:def 6;
    hence thesis;
  end;
  consider r being Element of L such that
A12: r = pow(x, i-j);
A13: len s = m by A7,FINSEQ_1:def 3;
  for k being Nat st 1 <= k & k <= len s holds s.k = (pow(x,i-j)) |^ (k-' 1)
  proof
    let k be Nat;
    assume that
A14: 1 <= k and
A15: k <= len s;
A16: 1 - 1 <= k - 1 by A14,XREAL_1:9;
    s.k = s/.k by A14,A15,FINSEQ_4:15
      .= pow(x, (i-j)*(k-1)) by A9,A13,A14,A15
      .= pow(x, (i-j)*(k-'1)) by A16,XREAL_0:def 2
      .= pow(pow(x,i-j),k-'1) by A4,Th26
      .= (pow(x,i-j)) |^ (k-'1) by Def2;
    hence thesis;
  end;
  then Sum s = (1.L-((pow(x, i-j)) |^ (len s)) ) / (1.L-pow(x, i-j)) by A1,A2
,A3,Th5,Th32
    .= (1.L-((pow(x, i-j)) |^ m))/(1.L-pow(x, i-j)) by A7,FINSEQ_1:def 3
    .= (1.L - (pow(pow(x,i-j),m))) / (1.L - pow(x,i-j)) by Def2
    .= (1.L - (pow(x,(i-j)*m))) / (1.L - pow(x,i-j)) by A4,Th26
    .= 0.L / (1.L - r) by A5,A12,VECTSP_1:19
    .= 0.L;
  hence thesis by A2,A3,A9,A13,Th37;
end;
