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 Th93:
  j in Seg len M9 & j<>i implies the_rank_of DelLine(M9,i) =
  the_rank_of RLine(M9,i,a*Line(M9,j))
proof
  assume that
A1: j in Seg len M9 and
A2: i<>j;
  per cases;
  suppose
A3: i in Seg len M9;
    set Li=Line(M9,i);
    set W=width M9;
    set R=RLine(M9,i,0.K*Li);
A4: W=len (0.K*Li) by CARD_1:def 7;
    then
A5: len R=len M9 by MATRIX11:def 3;
    set Lj=Line(M9,j);
A6: W=len (a*Lj) by CARD_1:def 7;
    reconsider 0Li=0.K*Li,aLj=a*Lj as Element of (the carrier of K)* by
FINSEQ_1:def 11;
    width R=W by A4,MATRIX11:def 3;
    then
A7: RLine(R,i,aLj) = Replace(R,i,aLj) by A6,MATRIX11:29
      .= Replace(Replace(M9,i,0Li),i,aLj) by A4,MATRIX11:29
      .= Replace(M9,i,aLj) by FUNCT_7:34
      .= RLine(M9,i,aLj) by A6,MATRIX11:29;
A8: len M9=n9 by MATRIX_0:def 2;
    then
A9: Line(R,j) =Line(M9,j) by A1,A2,MATRIX11:28;
    Line(R,i)=0.K*Li by A3,A4,A8,MATRIX11:28;
    then
A10: Line(R,i)+a*Line(R,j) = (W|->0.K)+a*Line(M9,j)by A9,FVSUM_1:58
      .= a*Line(M9,j) by FVSUM_1:21;
    W=len Li by CARD_1:def 7;
    hence the_rank_of DelLine(M9,i) = the_rank_of R by Th91
      .= the_rank_of RLine(M9,i,a*Lj)by A1,A2,A5,A10,A7,Th92;
  end;
  suppose
A11: not i in Seg len M9;
    then not i in dom M9 by FINSEQ_1:def 3;
    then DelLine(M9,i)=M9 by FINSEQ_3:104;
    hence thesis by A11,Th40;
  end;
end;
