reserve i,j,k,n,l for Nat,
  K for Field,
  a,b,c for Element of K,
  p,q for FinSequence of K,
  M1,M2,M3 for Matrix of n,K;
reserve D for non empty set,
  t for FinSequence of D,
  A for Matrix of n,D;

theorem Th6:
  for p being FinSequence of D holds
  0 < n & A is_symmetry_circulant_about p implies
  A@ is_symmetry_circulant_about p
  proof
    let p be FinSequence of D;
    assume that
A1: 0 < n and
A2: A is_symmetry_circulant_about p;
A3: len p=width A by A2;
  width A=n & len A=n by MATRIX_0:24;
  then
A4:  width (A@)=len p by A1,A3,MATRIX_0:54;
A5:for i,j be Nat st [i,j] in Indices A@ & i+j<>len p+1 holds A@*(i,j)=
     p.(i+j-1 mod len p)
  proof
    let i,j be Nat;
A6: Indices A = [:Seg n, Seg n:] by MATRIX_0:24;
    assume that
A7:[i,j] in Indices A@ and
A8:i+j<>len p+1;
    [i,j] in Indices A by A7,MATRIX_0:26;
    then i in Seg n & j in Seg n by A6,ZFMISC_1:87;
    then
A9: [j,i] in Indices A by A6,ZFMISC_1:87;
    then A@*(i,j) = A*(j,i) by MATRIX_0:def 6;
    hence thesis by A2,A9,A8;
  end;
   for i,j be Nat st [i,j] in Indices A@ & i+j=len p+1 holds A@*(i,j)=
     p.(len p)
    proof
      let i,j be Nat;
A10: Indices A = [:Seg n, Seg n:] by MATRIX_0:24;
      assume that
A11:[i,j] in Indices A@ and
A12:i+j=len p+1;
      [i,j] in Indices A by A11,MATRIX_0:26;
      then i in Seg n & j in Seg n by A10,ZFMISC_1:87;
      then
A13: [j,i] in Indices A by A10,ZFMISC_1:87;
      then A@*(i,j) = A*(j,i) by MATRIX_0:def 6;
      hence thesis by A2,A12,A13;
  end;
  hence thesis by A4,A5;
end;
