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 & q is first-col-of-circulant & len p =
  len q & len p > 0 implies a*(CCirc p)+a*(CCirc q)=CCirc(a*(p+q))
proof
  assume that
A1: p is first-col-of-circulant & q is first-col-of-circulant and
A2: len p = len q;
A3: len CCirc(p)= len p & width CCirc(p)=len p by MATRIX_0:24;
  len CCirc(q)= len p & width CCirc(q)=len p by A2,MATRIX_0:24;
  then a*(CCirc p)+a*(CCirc q) =a*(CCirc p+CCirc q) by A3,MATRIX_5:20
    .=a*(CCirc (p+q)) by A1,A2,Th38
    .=CCirc(a*(p+q)) by A1,A2,Th37,Th47;
  hence thesis;
end;
