reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;

theorem Th91:
  for p st len p = width M9 holds the_rank_of DelLine(M9,i) =
  the_rank_of RLine(M9,i,0.K*p)
proof
  let p such that
A1: len p = width M9;
  set R=RLine(M9,i,0.K*p);
A2: Seg len M9=dom M9 by FINSEQ_1:def 3;
A3: len M9=n9 by MATRIX_0:def 2;
  per cases;
  suppose
A4: not i in dom M9;
    then R=M9 by A2,Th40;
    hence thesis by A4,FINSEQ_3:104;
  end;
  suppose
A5: i in dom M9;
   then
A6: n9 <> 0 by A2,A3;
    set KK=the carrier of K;
A7: p is Element of (len p)-tuples_on KK by FINSEQ_2:92;
A8: len (0.K *p)=len p by MATRIXR1:16;
    then Line(R,i)=0.K * p by A1,A2,A3,A5,MATRIX11:28;
    then
A9: Line(R,i)=(len p) |-> 0.K by A7,FVSUM_1:58;
    reconsider 0p=0.K*p as Element of KK* by FINSEQ_1:def 11;
    R=Replace(M9,i,0p) by A1,A8,MATRIX11:29;
    then
A11: Replace(R,i,0p)=Replace(M9,i,0p) by FUNCT_7:34;
A12: width R=m9 by Th1,A6;
    width M9=m9 by Th1,A6;
    then the_rank_of R=the_rank_of DelLine(R,i) by A1,A12,A9,Th90;
    hence thesis by A11,COMPUT_1:4;
  end;
end;
