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 Th58:
  p is first-line-of-anti-circular implies -p is first-line-of-anti-circular
proof
  set n=len p;
  assume p is first-line-of-anti-circular;
  then consider M1 being Matrix of len p,K such that
A1: M1 is_anti-circular_about p;
  set M2=-M1;
A2: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (-M1) = [: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
A4: -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then
A5: len (-p)=len p by CARD_1:def 7;
  then
A6: dom (-p)=Seg len p by FINSEQ_1:def 3;
A7: for i,j be Nat st [i,j] in Indices (-M1)&i>=j holds (-M1)*(i,j)=(-(-p))
  .((j-i mod len (-p))+1)
  proof
    let i,j be Nat;
    assume that
A8: [i,j] in Indices (-M1) and
A9: i>=j;
A10: (j-i mod n)+1 in Seg n by A3,A8,Lm3;
    (-M1)*(i,j)=-(M1*(i,j)) by A2,A3,A8,MATRIX_3:def 2
      .=(comp K).(M1*(i,j)) by VECTSP_1:def 13
      .=(comp K).((-p).((j-i mod len p)+1) ) by A1,A2,A3,A8,A9
      .=(-(-p)).((j-i mod len p)+1) by A6,A10,FUNCT_1:13;
    hence thesis by A4,CARD_1:def 7;
  end;
A11: for i,j be Nat st [i,j] in Indices (-M1) & i<=j holds (-M1)*(i,j)=(-p).(
  (j-i mod len (-p))+1)
  proof
    let i,j be Nat;
    assume that
A12: [i,j] in Indices (-M1) and
A13: i<=j;
    (j-i mod n)+1 in Seg n by A3,A12,Lm3;
    then
A14: (j-i mod len p)+1 in dom p by FINSEQ_1:def 3;
    (-M1)*(i,j)=-(M1*(i,j)) by A2,A3,A12,MATRIX_3:def 2
      .=(comp K).(M1*(i,j)) by VECTSP_1:def 13
      .=(comp K).(p.((j-i mod len p)+1) ) by A1,A2,A3,A12,A13
      .=(-p).((j-i mod len p)+1) by A14,FUNCT_1:13;
    hence thesis by A4,CARD_1:def 7;
  end;
  width (-M1)=n by MATRIX_0:24;
  then M2 is_anti-circular_about -p by A5,A11,A7;
  then consider M2 being Matrix of len -p,K such that
A15: M2 is_anti-circular_about -p by A5;
  take M2;
  thus thesis by A15;
end;
