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
  A is line_circulant & [i,j] in [:Seg n, Seg n:] & k=i+1 & l=j+1 & i<n
  & j<n implies A*(i,j)=A*(k,l)
proof
  assume that
A1: A is line_circulant and
A2: [i,j] in [:Seg n, Seg n:] and
A3: k=i+1 and
A4: l=j+1 and
A5: i<n and
A6: j<n;
  consider p being FinSequence of D such that
  len p=width A and
A7: A is_line_circulant_about p by A1;
A8: Indices A=[:Seg n, Seg n:] by MATRIX_0:24;
  j in Seg n by A2,ZFMISC_1:87;
  then 1 <= j by FINSEQ_1:1;
  then 1+1 <= j+1 by XREAL_1:6;
  then
A9: 1 <= j+1 by XXREAL_0:2;
  j+1<=n by A6,INT_1:7;
  then
A10: l in Seg n by A4,A9;
  i in Seg n by A2,ZFMISC_1:87;
  then 1 <= i by FINSEQ_1:1;
  then 1+1 <= i+1 by XREAL_1:6;
  then
A11: 1 <= i+1 by XXREAL_0:2;
  i+1<=n by A5,INT_1:7;
  then k in Seg n by A3,A11;
  then [k,l] in Indices A by A8,A10,ZFMISC_1:87;
  then A*(k,l)=p.((l-k mod len p)+1) by A7
    .=p.((j-i mod len p)+1) by A3,A4
    .=A*(i,j) by A2,A7,A8;
  hence thesis;
end;
