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