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;
