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)+a*(ACirc q)=ACirc(a*(p+q))
proof
  assume that
A1: p is first-line-of-anti-circular & q is first-line-of-anti-circular and
A2: len p = len q;
A3: len ACirc(p)= len p & width ACirc(p) = len p by MATRIX_0:24;
  len ACirc(q)= len p & width ACirc(q) = len p by A2,MATRIX_0:24;
  then a*(ACirc p)+a*(ACirc q)=a*(ACirc p+ACirc q) by A3,MATRIX_5:20
    .=a*(ACirc (p+q)) by A1,A2,Th61
    .=ACirc(a*(p+q)) by A1,A2,Th60,Th63;
  hence thesis;
end;
