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 Th31:
  p is first-line-of-circulant implies -p is first-line-of-circulant
proof
  set n=len p;
  assume p is first-line-of-circulant;
  then consider M1 being Matrix of len p,K such that
A1: M1 is_line_circulant_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;
A6: dom p=Seg len p by FINSEQ_1:def 3;
A7: for i,j be Nat st [i,j] in Indices (-M1) holds (-M1)*(i,j)=(-p).((j-i
  mod len (-p))+1)
  proof
    let i,j be Nat;
    assume
A8: [i,j] in Indices (-M1);
    then
A9: (j-i mod n)+1 in Seg n by A3,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
      .=(-p).((j-i mod len p)+1) by A6,A9,FUNCT_1:13;
    hence thesis by A4,CARD_1:def 7;
  end;
  width (-M1)=n by MATRIX_0:24;
  then M2 is_line_circulant_about -p by A5,A7;
  then consider M2 being Matrix of len -p,K such that
A10: M2 is_line_circulant_about -p by A5;
  take M2;
  thus thesis by A10;
end;
