reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;
reserve x,y,x1,x2,y1,y2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of D;

theorem
 not i in Seg width M implies DelCol(M,i) = M
 proof assume
A1: not i in Seg width M;
A2: len DelCol(M,i) = len M by Def13;
  for k st 1 <= k & k <= len M holds M.k = DelCol(M,i).k
   proof let k such that
A3: 1 <= k & k <= len M;
A4:   k in dom M by A3,FINSEQ_3:25;
      len Line(M,k) = width M by Def7;
      then
A5:   not i in dom Line(M,k) by A1,FINSEQ_1:def 3;
      thus M.k = Line(M,k) by A4,Lm1
        .= Del(Line(M,k),i) by A5,FINSEQ_3:104
        .= DelCol(M,i).k by A4,Def13;
   end;
 hence DelCol(M,i) = M by A2,FINSEQ_1:14;
 end;
