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 ACirc(-p) =-(ACirc(p))
proof
  set n=len p;
A1: len ACirc(p)= len p & width ACirc(p) = len p by MATRIX_0:24;
A2: Indices ACirc(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-anti-circular;
  then -p is first-line-of-anti-circular by Th58;
  then
A5: ACirc(-p) is_anti-circular_about -p by Def12;
A6: ACirc(p) is_anti-circular_about p by A4,Def12;
A7: for i,j be Nat st [i,j] in Indices ACirc(p) holds ACirc(-p)*(i,j)=-(
  ACirc(p)*(i,j))
  proof
    let i,j be Nat;
    assume
A8: [i,j] in Indices ACirc(p);
    now
      per cases;
      case
A9:     i<=j;
        (j-i mod n)+1 in Seg n by A2,A8,Lm3;
        then
A10:    (j-i mod len p)+1 in dom p by FINSEQ_1:def 3;
        [i,j] in Indices ACirc(-p) by A3,A8,MATRIX_0:26;
        then ACirc(-p)*(i,j)=(-p).((j-i mod len (-p))+1) by A5,A9
          .=(comp K).(p.((j-i mod len p)+1)) by A3,A10,FUNCT_1:13
          .=(comp K).(ACirc(p)*(i,j)) by A6,A8,A9
          .=-(ACirc(p)*(i,j)) by VECTSP_1:def 13;
        hence thesis;
      end;
      case
A11:    i>=j;
        (j-i mod n)+1 in Seg n by A2,A8,Lm3;
        then
A12:    (j-i mod len p)+1 in dom -p by A3,FINSEQ_1:def 3;
        [i,j] in Indices ACirc(-p) by A3,A8,MATRIX_0:26;
        then ACirc(-p)*(i,j) =(-(-p)).((j-i mod len (-p))+1) by A5,A11
          .=(comp K).((-p).((j-i mod len p)+1)) by A3,A12,FUNCT_1:13
          .=(comp K).(ACirc(p)*(i,j)) by A6,A8,A11
          .=-(ACirc(p)*(i,j)) by VECTSP_1:def 13;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  len ACirc(-p)= len p & width ACirc(-p) = len p by A3,MATRIX_0:24;
  hence thesis by A1,A7,MATRIX_3:def 2;
end;
