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 Th15:
  for f be FinSequence of K, i,j st i in Seg n & j in Seg n holds
  Delete(M,i,j) = Delete(RLine(M,i,f),i,j)
proof
  let f be FinSequence of K, i,j such that
A1: i in Seg n and
A2: j in Seg n;
A3: Delete(M,i,j) = Deleting(M,i,j) by A1,A2,Def1;
A4: Delete(RLine(M,i,f),i,j) = Deleting(RLine(M,i,f),i,j) by A1,A2,Def1;
  reconsider f9=f as Element of (the carrier of K)* by FINSEQ_1:def 11;
  reconsider I=i as Element of NAT by ORDINAL1:def 12;
  per cases;
  suppose
    len f=width M;
    then RLine(M,I,f)=Replace(M,i,f9) by MATRIX11:29;
    hence thesis by A3,A4,COMPUT_1:3;
  end;
  suppose
    len f<>width M;
    hence thesis by MATRIX11:def 3;
  end;
end;
