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;

theorem Th2:
  (n,n)-->a is_symmetry_circulant_about n|->a
  proof
    set p=n|->a;
    set M=(n,n)-->a;
A1: width M=n & len p =n by CARD_1:def 7,MATRIX_0:24;
    hence len p = width M;
A2: Indices ((n,n)-->a)=[:Seg n, Seg n:] by MATRIX_0:24;
    thus for i,j be Nat st [i,j] in Indices M & i+j<>len p+1 holds
    M*(i,j)=p.((i+j-1) mod len p)
    proof
      let i,j be Nat;
      assume that
A3:   [i,j] in Indices M and
A4:   i+j<>len p+1;
      (Seg n --> a).(i+j-1 mod len p)=a by A1,A2,A3,A4,Lm4,FUNCOP_1:7;
      hence thesis by A3,MATRIX_0:46;
    end;
    let i,j be Nat;
     assume that
A5:[i,j] in Indices ((n,n)-->a) and
A6:i+j=len p+1;
       i in Seg n & j in Seg n by A2,A5,ZFMISC_1:87;
       then
        1<=i & 1<=j by FINSEQ_1:1;
       then
        1+1<=i+j by XREAL_1:7;
       then len p +1-1 >=1+1-1 by A6,XREAL_1:9;
       then
A7:    len p in Seg len p;
      (Seg n --> a).(len p)=a by A1,A7,FUNCOP_1:7;
      hence thesis by A5,MATRIX_0:46;
   end;
