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;
