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 Th10:
  p is first-symmetry-of-circulant & q is first-symmetry-of-circulant &
  len p=len q implies p+q is first-symmetry-of-circulant
proof
  set n = len p;
  assume that
A1: p is first-symmetry-of-circulant and
A2: q is first-symmetry-of-circulant and
A3: len p=len q;
  consider M2 being Matrix of n,K such that
A4: M2 is_symmetry_circulant_about q by A2,A3;
A5: dom p=Seg n by FINSEQ_1:def 3;
  dom q=Seg n by A3,FINSEQ_1:def 3;
  then dom (p+q)=dom p by A5,POLYNOM1:1; then
A6: len (p+q)=n by A5,FINSEQ_1:def 3;
  consider M1 being Matrix of n,K such that
A7: M1 is_symmetry_circulant_about p by A1;
  width (M1+M2)=n by MATRIX_0:24;
  then consider M3 being Matrix of len (p+q),K such that
  len (p+q)=width M3 and
A8: M3 is_symmetry_circulant_about p+q by A6,A4,A7,Th5;
  take M3;
  thus thesis by A8;
end;
