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
  for p being FinSequence of K st p is first-col-of-circulant holds
  CCirc(-p) =-(CCirc p)
proof
  let p;
  set n = len p;
A1: dom p=Seg len p by FINSEQ_1:def 3;
A2: Indices CCirc(p) =[:Seg n, Seg n:] by MATRIX_0:24;
  assume
A3: p is first-col-of-circulant;
  then
A4: CCirc(p) is_col_circulant_about p by Def8;
  -p is first-col-of-circulant by A3,Th35;
  then
A5: CCirc(-p) is_col_circulant_about -p by Def8;
  p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  then -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then
A6: len (-p)=len p by CARD_1:def 7;
  then
A7: Indices CCirc(p) =Indices CCirc(-p) by MATRIX_0:26;
A8: for i,j be Nat st [i,j] in Indices CCirc(p) holds CCirc(-p)*(i,j)=-(
  CCirc(p)*(i,j))
  proof
    let i,j be Nat;
    assume
A9: [i,j] in Indices CCirc(p);
    then
A10: (i-j mod n)+1 in Seg n by A2,Lm3;
    CCirc(-p)*(i,j) =(-p).((i-j mod len -p)+1) by A5,A7,A9
      .=(comp K).(p.((i-j mod len p)+1)) by A6,A1,A10,FUNCT_1:13
      .=(comp K).(CCirc(p)*(i,j)) by A4,A9
      .= -(CCirc(p)*(i,j)) by VECTSP_1:def 13;
    hence thesis;
  end;
A11: len CCirc(p)= len p & width CCirc(p) = len p by MATRIX_0:24;
  len CCirc(-p)= len p & width CCirc(-p) = len p by A6,MATRIX_0:24;
  hence thesis by A11,A8,MATRIX_3:def 2;
end;
