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 Th11:
  M1 is line_circulant implies -M1 is line_circulant
proof
A1: width M1=n by MATRIX_0:24;
A2: Indices (-M1) = [:Seg n, Seg n:] by MATRIX_0:24;
  assume M1 is line_circulant;
  then consider p being FinSequence of K such that
A3: len p=width M1 and
A4: M1 is_line_circulant_about p;
  p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  then
A5: -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then
A6: width (-M1)=n & len (-p)=len p by CARD_1:def 7,MATRIX_0:24;
A7: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
  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 (j-i mod n)+1 in Seg n by A2,Lm3;
    then
A9: (j-i mod len p)+1 in dom p by A3,A1,FINSEQ_1:def 3;
    (-M1)*(i,j) =-(M1*(i,j)) by A7,A2,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 A4,A7,A2,A8
      .=(-p).((j-i mod len p)+1) by A9,FUNCT_1:13;
    hence thesis by A5,CARD_1:def 7;
  end;
  then -M1 is_line_circulant_about -p by A3,A1,A6;
  then consider r being FinSequence of K such that
A10: len r =width (-M1) & -M1 is_line_circulant_about r;
  take r;
  thus thesis by A10;
end;
