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 Th17:
  M1 is symmetry_circulant implies M1@ = M1
  proof
    assume M1 is symmetry_circulant;
    then consider p being FinSequence of K such that len p=width M1 and
A1: M1 is_symmetry_circulant_about p;
A2:Indices M1 = [:Seg n,Seg n:] by MATRIX_0:24;
A3:len M1=n & width M1=n by MATRIX_0:24;
A4:len (M1@) = n & width (M1@) = n by MATRIX_0:24;
   for i,j be Nat st [i,j] in Indices M1 holds
   M1@*(i,j)=M1*(i,j)
   proof
    let i,j be Nat;
    assume
A5:[i,j] in Indices M1;
     per cases;
     suppose
A6:i+j<>len p +1;
     i in Seg n & j in Seg n by A2,A5,ZFMISC_1:87;
     then
A7:[j,i] in [:Seg n,Seg n:] by ZFMISC_1:87;
     then
     M1@*(i,j)=M1*(j,i) by A2,MATRIX_0:def 6
     .=p.(i+j-1 mod len p) by A1,A7,A6,A2;
     hence thesis by A1,A5,A6;
   end;
     suppose
A8:i+j=len p+1;
     i in Seg n & j in Seg n by A5,A2,ZFMISC_1:87;
     then
A9:  [j,i] in [:Seg n,Seg n:] by ZFMISC_1:87;
     M1@*(i,j)=M1*(j,i) by A2,A9,MATRIX_0:def 6
       .=p.(len p) by A1,A9,A8,A2;
     hence thesis by A1,A8,A5;
    end;
   end;
   hence thesis by A3,A4,MATRIX_0:21;
  end;
