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-circulant implies LCirc(-p) =-(LCirc p)
proof
  set n=len p;
A1: len LCirc(p)= len p & width LCirc(p)=len p by MATRIX_0:24;
A2: Indices LCirc(p) =[:Seg n, Seg n:] by MATRIX_0:24;
  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
A3: len (-p)=len p by CARD_1:def 7;
  assume
A4: p is first-line-of-circulant;
  then -p is first-line-of-circulant by Th31;
  then
A5: LCirc(-p) is_line_circulant_about -p by Def7;
A6: LCirc(p) is_line_circulant_about p by A4,Def7;
A7: for i,j be Nat st [i,j] in Indices LCirc(p) holds LCirc(-p)*(i,j)=-(
  LCirc(p)*(i,j))
  proof
    let i,j be Nat;
    assume
A8: [i,j] in Indices LCirc(p);
    then (j-i mod n)+1 in Seg n by A2,Lm3;
    then
A9: (j-i mod len p)+1 in dom p by FINSEQ_1:def 3;
    [i,j] in Indices LCirc(-p) by A3,A8,MATRIX_0:26;
    then LCirc(-p)*(i,j) =(-p).((j-i mod len -p)+1) by A5
      .=(comp K).( p.((j-i mod len p)+1)) by A3,A9,FUNCT_1:13
      .=(comp K).(LCirc(p)*(i,j)) by A6,A8
      .=-(LCirc(p)*(i,j)) by VECTSP_1:def 13;
    hence thesis;
  end;
  len LCirc(-p)= len p & width LCirc(-p)=len p by A3,MATRIX_0:24;
  hence thesis by A1,A7,MATRIX_3:def 2;
end;
