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 implies a*(ACirc p)+b*(ACirc p)=ACirc
  ((a+b)*p)
proof
A1: len (b*p)=len p by MATRIXR1:16;
A2: p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  assume
A3: p is first-line-of-anti-circular;
  then
A4: a*p is first-line-of-anti-circular & b*p is first-line-of-anti-circular
  by Th62;
  a*(ACirc p)+b*(ACirc p)=ACirc (a*p)+b*(ACirc p) by A3,Th63
    .=ACirc (a*p)+ACirc (b*p) by A3,Th63
    .=ACirc(a*p+b*p) by A4,A1,Th61,MATRIXR1:16;
  hence thesis by A2,FVSUM_1:55;
end;
