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 Th8:
  p is first-symmetry-of-circulant implies -p is first-symmetry-of-circulant
proof
  set n=len p;
  assume p is first-symmetry-of-circulant;
  then consider M1 being Matrix of len p,K such that
A1: M1 is_symmetry_circulant_about p;
  p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  then -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then
A2: len (-p)=len p by CARD_1:def 7;
  -M1 is_symmetry_circulant_about -p by A1,Th4;
  then consider M2 being Matrix of len -p,K such that
A3: M2 is_symmetry_circulant_about -p by A2;
  take M2;
  thus thesis by A3;
end;
