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 implies a*(CCirc p)+b*(CCirc q)=CCirc(a*p+b*q)
proof
  set n = len p;
  assume that
A1: p is first-col-of-circulant and
A2: q is first-col-of-circulant and
A3: len p = len q;
A4: a*p is first-col-of-circulant & b*q is first-col-of-circulant by A1,A2,Th46
;
A5: len (b*q)=n by A3,MATRIXR1:16;
  a*(CCirc p)+b*(CCirc q)=CCirc (a*p)+b*(CCirc q) by A1,Th47
    .=CCirc (a*p)+CCirc (b*q) by A2,Th47
    .=CCirc(a*p+b*q) by A4,A5,Th38,MATRIXR1:16;
  hence thesis;
end;
