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
  M1 is anti-circular & [i,j] in [:Seg n, Seg n:] & k=i+1 & l=j+1 & i<n
  & j<n implies M1*(k,l)=M1*(i,j)
proof
  now
    per cases;
    case
A1:   i<=j;
      then
A2:   i+1<= j+1 by XREAL_1:6;
A3:   Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
      assume that
A4:   M1 is anti-circular and
A5:   [i,j] in [:Seg n, Seg n:] and
A6:   k=i+1 and
A7:   l=j+1 and
A8:   i<n and
A9:   j<n;
      j in Seg n by A5,ZFMISC_1:87;
      then 1 <= j by FINSEQ_1:1;
      then 1+1 <= j+1 by XREAL_1:6;
      then
A10:  1 <= j+1 by XXREAL_0:2;
      j+1<=n by A9,INT_1:7;
      then
A11:  l in Seg n by A7,A10;
      i in Seg n by A5,ZFMISC_1:87;
      then 1 <= i by FINSEQ_1:1;
      then 1+1 <= i+1 by XREAL_1:6;
      then
A12:  1 <= i+1 by XXREAL_0:2;
      consider p being FinSequence of K such that
      len p=width M1 and
A13:  M1 is_anti-circular_about p by A4;
      i+1<=n by A8,INT_1:7;
      then k in Seg n by A6,A12;
      then [k,l] in Indices M1 by A3,A11,ZFMISC_1:87;
      then M1*(k,l)=p.((l-k mod len p)+1) by A6,A7,A13,A2
        .=p.((j-i mod len p)+1) by A6,A7
        .=M1*(i,j) by A1,A5,A13,A3;
      hence thesis;
    end;
    case
A14:  i>=j;
      then
A15:  i+1>=j+1 by XREAL_1:6;
A16:  Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
      assume that
A17:  M1 is anti-circular and
A18:  [i,j] in [:Seg n, Seg n:] and
A19:  k=i+1 and
A20:  l=j+1 and
A21:  i<n and
A22:  j<n;
      j in Seg n by A18,ZFMISC_1:87;
      then 1 <= j by FINSEQ_1:1;
      then 1+1<=j+1 by XREAL_1:6;
      then
A23:  1<=j+1 by XXREAL_0:2;
      j+1<=n by A22,INT_1:7;
      then
A24:  l in Seg n by A20,A23;
      i in Seg n by A18,ZFMISC_1:87;
      then 1 <= i by FINSEQ_1:1;
      then 1+1<=i+1 by XREAL_1:6;
      then
A25:  1<=i+1 by XXREAL_0:2;
      consider p being FinSequence of K such that
      len p=width M1 and
A26:  M1 is_anti-circular_about p by A17;
      i+1<=n by A21,INT_1:7;
      then k in Seg n by A19,A25;
      then [k,l] in Indices M1 by A16,A24,ZFMISC_1:87;
      then M1*(k,l)=(-p).((l-k mod len p)+1) by A19,A20,A26,A15
        .=(-p).((j-i mod len p)+1) by A19,A20
        .=M1*(i,j) by A14,A18,A26,A16;
      hence thesis;
    end;
  end;
  hence thesis;
end;
