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-line-of-circulant & q is first-line-of-circulant & len p =
  len q implies a*(LCirc p)+a*(LCirc q)=LCirc(a*(p+q))
proof
  assume that
A1: p is first-line-of-circulant & q is first-line-of-circulant and
A2: len p=len q;
A3: len LCirc(p)= len p & width LCirc(p) = len p by MATRIX_0:24;
  len LCirc(q)= len p & width LCirc(q)=len p by A2,MATRIX_0:24;
  then a*(LCirc p)+a*(LCirc q)=a*(LCirc p+LCirc q) by A3,MATRIX_5:20
    .=a*(LCirc (p+q)) by A1,A2,Th34
    .=LCirc(a*(p+q)) by A1,A2,Th33,Th42;
  hence thesis;
end;
