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 Th12:
  p is first-symmetry-of-circulant implies a*p is first-symmetry-of-circulant
proof
  set n=len p;
A1: len (a*p)=len p by MATRIXR1:16;
  assume p is first-symmetry-of-circulant;
  then consider M1 being Matrix of n,K such that
A2: M1 is_symmetry_circulant_about p;
  a*M1 is_symmetry_circulant_about a*p by A2,Th3;
  then consider M2 being Matrix of len (a*p),K such that
A3: M2 is_symmetry_circulant_about a*p by A1;
  take M2;
  thus thesis by A3;
end;
