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 implies
  a*(SCirc p)+b*(SCirc p) = SCirc((a+ b)*p)
proof
A1: len (b*p)=len p by MATRIXR1:16;
A2: p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  assume
A3: p is first-symmetry-of-circulant;
  then
A4: a*p is first-symmetry-of-circulant & b*p is first-symmetry-of-circulant
  by Th12;
  a*(SCirc p)=SCirc(a*p) & b*(SCirc p)=SCirc(b*p) by A3,Th13;
  then a*(SCirc p)+b*(SCirc p)=SCirc(a*p+b*p) by A4,A1,Th11,MATRIXR1:16;
  hence thesis by A2,FVSUM_1:55;
end;
