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;
reserve D for non empty set,
  t for FinSequence of D,
  A for Matrix of n,D;

theorem
  p is first-symmetry-of-circulant implies SCirc(-p) = -(SCirc p)
proof
  set n=len p;
A1: len SCirc(p)= len p & width SCirc(p)=len p by MATRIX_0:24;
A2: Indices SCirc(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-symmetry-of-circulant;
  then -p is first-symmetry-of-circulant by Th8;
  then
A5: SCirc(-p) is_symmetry_circulant_about -p by Def7;
A6: SCirc(p) is_symmetry_circulant_about p by A4,Def7;
A7: for i,j be Nat st [i,j] in Indices SCirc(p) holds SCirc(-p)*(i,j)=-(
  SCirc(p)*(i,j))
  proof
    let i,j be Nat;
    assume
A8: [i,j] in Indices SCirc(p);
    now
      per cases;
      suppose
A9:  i+j<>len p +1;
       i+j-1 mod n in Seg n by A2,A8,A9,Lm4;
       then
A10: i+j-1 mod len p in dom p by FINSEQ_1:def 3;
       [i,j] in Indices SCirc(-p) by A3,A8,MATRIX_0:26;
       then SCirc(-p)*(i,j) =(-p).(i+j-1 mod len -p) by A5,A9,A3
       .=(comp K).( p.(i+j-1 mod len p)) by A3,A10,FUNCT_1:13
       .=(comp K).(SCirc(p)*(i,j)) by A6,A8,A9
       .=-(SCirc(p)*(i,j)) by VECTSP_1:def 13;
       hence thesis;
     end;
     suppose
A11: i+j=len p+1;
       i in Seg n & j in Seg n by A2,A8,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 A11,XREAL_1:9;
       then len p in Seg len p;
       then
A12:len p in dom p by FINSEQ_1:def 3;
       [i,j] in Indices SCirc(-p) by A3,A8,MATRIX_0:26;
       then SCirc(-p)*(i,j) =(-p).(len -p) by A5,A11,A3
       .=(comp K).( p.(len p)) by A3,A12,FUNCT_1:13
       .=(comp K).(SCirc(p)*(i,j)) by A6,A8,A11
       .=-(SCirc(p)*(i,j)) by VECTSP_1:def 13;
       hence thesis;
      end;
    end;
    hence thesis;
   end;
  len SCirc(-p)= len p & width SCirc(-p)=len p by A3,MATRIX_0:24;
  hence thesis by A1,A7,MATRIX_3:def 2;
end;
