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-anti-circular & q is first-line-of-anti-circular &
  len p = len q implies a*(ACirc p)+b*(ACirc q)=ACirc(a*p+b*q)
proof
  set n = len p;
  assume that
A1: p is first-line-of-anti-circular and
A2: q is first-line-of-anti-circular and
A3: len p = len q;
A4: a*p is first-line-of-anti-circular & b*q is first-line-of-anti-circular
  by A1,A2,Th62;
A5: len (b*q)=n by A3,MATRIXR1:16;
  a*(ACirc p)+b*(ACirc q)=ACirc (a*p)+b*(ACirc q) by A1,Th63
    .=ACirc (a*p)+ACirc (b*q) by A2,Th63
    .=ACirc(a*p+b*q) by A4,A5,Th61,MATRIXR1:16;
  hence thesis;
end;
