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 Th90:
  Line(M,i)=(width M) |-> 0.K implies the_rank_of DelLine(M,i) = the_rank_of M
proof
  set D=DelLine(M,i);
A1: Indices M=[:Seg len M,Seg width M:] by FINSEQ_1:def 3;
A2: Segm(M,Seg len M \ {i},Seg width M) = D by Th51;
  consider P,Q such that
A3: [:P,Q:] c= Indices D and
A4: card P = card Q and
A5: card P = the_rank_of D and
A6: Det EqSegm(D,P,Q)<>0.K by Def4;
  EqSegm(D,P,Q)=Segm(D,P,Q) by A4,Def3;
  then
A7: the_rank_of Segm(D,P,Q)=card P by A6,Th83;
  P={} iff Q={} by A4;
  then consider P2,Q2 such that
A8: P2 c= Seg len M \ {i} and
A9: Q2 c= Seg width M and
  P2 = Sgm (Seg len M \ {i}).:P and
  Q2=Sgm (Seg width M).:Q and
  card P2=card P and
  card Q2=card Q and
A10: Segm(D,P,Q) = Segm(M,P2,Q2) by A3,A2,Th57;
  Seg len M \ {i} c= Seg len M by XBOOLE_1:36;
  then P2 c= Seg len M by A8;
  then [:P2,Q2:] c= Indices M by A9,A1,ZFMISC_1:96;
  then
A11: the_rank_of D <= the_rank_of M by A5,A10,A7,Th79;
  consider p,q be without_zero finite Subset of NAT such that
A12: [:p,q:] c= Indices M and
A13: card p = card q and
A14: card p = the_rank_of M and
A15: Det EqSegm(M,p,q)<>0.K by Def4;
  EqSegm(M,p,q)=Segm(M,p,q) by A13,Def3;
  then
A16: the_rank_of Segm(M,p,q)=card p by A15,Th83;
  assume
A17: Line(M,i)=(width M) |-> 0.K;
  not i in p
  proof
    assume
A18: i in p;
    then reconsider i0=i as non zero Element of NAT;
    {i} c= p by A18,ZFMISC_1:31;
    then consider q1 be without_zero finite Subset of NAT such that
A19: q1 c= q and
A20: card {i} = card q1 and
A21: Det EqSegm(M,{i0},q1) <> 0.K by A13,A15,Th65;
    consider y being object such that
A22: {y}=q1 by A20,CARD_2:42;
A23: card { i } = 1 by CARD_1:30;
A24: q c= Seg width M by A12,A13,Th67;
    y in {y} by TARSKI:def 1;
    then reconsider y as non zero Element of NAT by A22;
    y in q1 by A22,TARSKI:def 1;
    then
A25: y in q by A19;
    then
A26: M*(i0,y)=Line(M,i).y by A24,MATRIX_0:def 7;
A27: Line(M,i).y =0.K by A17,A25,A24,FINSEQ_2:57;
    EqSegm(M,{i0},q1) = Segm(M,{i0},{y}) by A20,A22,Def3
      .= <*<* 0.K *>*> by A26,A27,Th44;
    hence thesis by A21,A23,MATRIX_3:34;
  end;
  then
A28: p\{i}=p by ZFMISC_1:57;
  p c= Seg len M by A12,A13,Th67;
  then
A29: p c= Seg len M\{i} by A28,XBOOLE_1:33;
  q c= Seg width M by A12,A13,Th67;
  then the_rank_of Segm(M,p,q)<=the_rank_of D by A2,A29,Th80;
  hence thesis by A11,A14,A16,XXREAL_0:1;
end;
