reserve x,y for object,
  N for Element of NAT,
  c,i,j,k,m,n for Nat,
  D for non empty set,
  s for Element of 2Set Seg (n+2),
  p for Element of Permutations(n) ,
  p1, q1 for Element of Permutations(n+1),
  p2 for Element of Permutations(n +2),
  K for Field,
  a for Element of K,
  f for FinSequence of K,
  A for (Matrix of K),
  AD for Matrix of n,m,D,
  pD for FinSequence of D,
  M for Matrix of n,K;

theorem
  not c in Seg width AD implies RCol(AD,c,pD) = AD
proof
  assume
A1: not c in Seg width AD;
  set R=RCol(AD,c,pD);
  per cases;
  suppose
A2: len pD=len AD;
    now
      let i,j be Nat such that
A3:   [i,j] in Indices AD;
      j in Seg width AD by A3,ZFMISC_1:87;
      hence R*(i,j) = AD*(i,j) by A1,A2,A3,Def2;
    end;
    hence thesis by MATRIX_0:27;
  end;
  suppose
    len pD<>len AD;
    hence thesis by Def2;
  end;
end;
