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
  M1 is line_circulant & M2 is line_circulant & M3 is line_circulant
  implies a*M1+b*M2-c*M3 is line_circulant
proof
  assume that
A1: M1 is line_circulant & M2 is line_circulant and
A2: M3 is line_circulant;
  c*M3 is line_circulant by A2,Th6;
  then
A3: -(c*M3) is line_circulant by Th11;
  a*M1 is line_circulant & b*M2 is line_circulant by A1,Th6;
  then a*M1 + b*M2 is line_circulant by Th7;
  hence thesis by A3,Th7;
end;
