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
  for K being Fanoian Field, n,i,j being Nat, M1 being Matrix of n,K st
  [i,j] in Indices M1 & i=j & M1 is anti-circular holds M1*(i,j)=0.K
proof
  let K be Fanoian Field;
  let n,i,j be Nat;
  let M1 be Matrix of n,K;
  assume that
A1: [i,j] in Indices M1 and
A2: i=j and
A3: M1 is anti-circular;
  consider p being FinSequence of K such that
A4: len p=width M1 and
A5: M1 is_anti-circular_about p by A3;
A6: M1*(i,j)=p.((j-i mod len p)+1) by A1,A2,A5;
A7: width M1=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 len (-p)=len p by CARD_1:def 7;
  then Indices M1=[:Seg n, Seg n:] & dom -p=Seg n by A4,A7,FINSEQ_1:def 3
,MATRIX_0:24;
  then
A8: (j-i mod len p)+1 in dom -p by A1,A4,A7,Lm3;
  M1*(i,j) =(-p).((j-i mod len p)+1) by A1,A2,A5
    .=(comp K).(p.((j-i mod len p)+1)) by A8,FUNCT_1:12
    .=-M1*(i,j) by A6,VECTSP_1:def 13;
  then M1*(i,j)+M1*(i,j)=0.K by RLVECT_1:5;
  then (1_K)*(M1*(i,j))+M1*(i,j)=0.K;
  then (1_K)*(M1*(i,j))+(1_K)*(M1*(i,j))=0.K;
  then 1_K+1_K<>0.K & (1_K+1_K)*(M1*(i,j))=0.K by VECTSP_1:def 7,def 19;
  hence thesis by VECTSP_1:12;
end;
