reserve i,j,k,n,l for 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;

theorem Th4:
  M1 is_symmetry_circulant_about p implies -M1 is_symmetry_circulant_about -p
  proof
    assume
A1: M1 is_symmetry_circulant_about p; then
A2: len p=width M1;
A3: width M1=n by MATRIX_0:24;
A4: 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
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;
A8:for i,j be Nat st [i,j] in Indices (-M1) & i+j<>len p+1 holds (-M1)*(i,j)=
(-p).(i+j-1 mod len (-p))
  proof
    let i,j be Nat;
    assume that
A9: [i,j] in Indices (-M1) and
A10:i+j<>len p+1;
    i+j-1 mod n in Seg n by A3,A2,A4,A9,A10,Lm4;
    then
A11: i+j-1 mod len p in dom p by A2,A3,FINSEQ_1:def 3;
    (-M1)*(i,j) =-(M1*(i,j)) by A7,A4,A9,MATRIX_3:def 2
      .=(comp K).(M1*(i,j)) by VECTSP_1:def 13
      .=(comp K).( p.(i+j-1 mod len p) ) by A1,A7,A4,A9,A10
      .=(-p).(i+j-1 mod len p) by A11,FUNCT_1:13;
    hence thesis by A5,CARD_1:def 7;
  end;
 for i,j be Nat st [i,j] in Indices (-M1) & i+j=len p+1 holds (-M1)*(i,j)=
(-p).(len (-p))
  proof
    let i,j be Nat;
    assume that
A12:[i,j] in Indices (-M1) and
A13:i+j=len p+1;
        i in Seg n & j in Seg n by A4,A12,ZFMISC_1:87;
       then
       1<=i & 1<=j by FINSEQ_1:1;
       then 1+1<=i+j by XREAL_1:7;
       then len p +1-1 >=1+1-1 by A13,XREAL_1:9;
       then len p in Seg len p;
       then
A14:  len p in dom p by FINSEQ_1:def 3;
     (-M1)*(i,j) =-(M1*(i,j)) by A7,A4,A12,MATRIX_3:def 2
      .=(comp K).(M1*(i,j)) by VECTSP_1:def 13
      .=(comp K).( p.(len p) ) by A1,A7,A4,A12,A13
      .=(-p).(len p) by A14,FUNCT_1:13;
    hence thesis by A5,CARD_1:def 7;
  end;
  hence thesis by A2,A3,A6,A8;
end;
