reserve i,j,k,n,l for Element of 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-col-of-circulant implies a*(CCirc p)+b*(CCirc p)=CCirc((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-col-of-circulant;
  then
A4: a*p is first-col-of-circulant & b*p is first-col-of-circulant by Th46;
  a*(CCirc p)=CCirc(a*p) & b*(CCirc p)=CCirc(b*p) by A3,Th47;
  then a*(CCirc p)+b*(CCirc p)=CCirc(a*p+b*p) by A4,A1,Th38,MATRIXR1:16;
  hence thesis by A2,FVSUM_1:55;
end;
