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;
