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
  p is first-symmetry-of-circulant & q is first-symmetry-of-circulant & len p =
  len q implies a*(SCirc p)+a*(SCirc q)=SCirc(a*(p+q))
proof
  assume that
A1: p is first-symmetry-of-circulant & q is first-symmetry-of-circulant and
A2: len p=len q;
A3: len SCirc(p)= len p & width SCirc(p) = len p by MATRIX_0:24;
  len SCirc(q)= len p & width SCirc(q)=len p by A2,MATRIX_0:24;
  then a*(SCirc p)+a*(SCirc q)=a*(SCirc p+SCirc q) by A3,MATRIX_5:20
    .=a*(SCirc (p+q)) by A1,A2,Th11
    .=SCirc(a*(p+q)) by A1,A2,Th10,Th13;
  hence thesis;
end;
