reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);

theorem Th10:
  for P,Q,Q9 be without_zero finite Subset of NAT st [:P,Q9:] c=
Indices A for i,j st i in dom A \ P & j in Seg width A \ Q & A*(i,j) <> 0.K & Q
c= Q9 & Line(A,i) * Sgm Q9 = card Q9 |-> 0.K holds the_rank_of A > the_rank_of
  Segm(A,P,Q)
proof
  let P,Q,R be without_zero finite Subset of NAT such that
A1: [:P,R:] c= Indices A;
  let i,j such that
A2: i in dom A \ P and
A3: j in Seg width A \ Q and
A4: A*(i,j)<> 0.K and
A5: Q c= R and
A6: Line(A,i) * Sgm R = card R |-> 0.K;
A7: dom A=Seg len A by FINSEQ_1:def 3;
  then
A8: i in Seg len A by A2,XBOOLE_0:def 5;
A9: [:P,Q:] c= [:P,R:] by A5,ZFMISC_1:95;
  then
A10: [:P,Q:] c= Indices A by A1;
  reconsider i0=i,j0=j as non zero Element of NAT by A2,A3,A7;
A11: j in Seg width A by A3,XBOOLE_0:def 5;
  set S=Segm(A,P,Q);
  consider P9,Q9 be without_zero finite Subset of NAT such that
A12: [:P9,Q9:] c= Indices S and
A13: card P9 = card Q9 and
A14: card P9 = the_rank_of S and
A15: Det EqSegm(S,P9,Q9)<>0.K by MATRIX13:def 4;
  P9={} iff Q9={} by A13;
  then consider P2,Q2 be without_zero finite Subset of NAT such that
A16: P2 c= P and
A17: Q2 c= Q and
  P2 = Sgm P.:P9 and
  Q2=Sgm Q.:Q9 and
A18: card P2=card P9 and
A19: card Q2=card Q9 and
A20: Segm(S,P9,Q9) = Segm(A,P2,Q2) by A12,MATRIX13:57;
  set Q2j=Q2\/{j0};
  set P2i=P2\/{i0};
  set ESS=EqSegm(A,P2i,Q2j);
  set SS=Segm(A,P2i,Q2j);
  per cases;
  suppose
    [:P,Q:] ={};
    then card P=0 or card Q=0 by CARD_1:27,ZFMISC_1:90;
    then
A21: the_rank_of S=0 by MATRIX13:77;
    [i,j] in Indices A by A7,A8,A11,ZFMISC_1:87;
    hence thesis by A4,A21,MATRIX13:94;
  end;
  suppose
A22: [:P,Q:]<>{};
    then P c= dom A by A10,ZFMISC_1:114;
    then
A23: P2 c= dom A by A16;
    [:P,R:]<>{} by A9,A22,XBOOLE_1:3;
    then
A24: R c= Seg width A by A1,ZFMISC_1:114;
A25: dom Sgm R=Seg card R by FINSEQ_3:40;
    Q c= Seg width A by A10,A22,ZFMISC_1:114;
    then
A26: Q2 c= Seg width A by A17;
A27: {j0}c= Seg width A by A11,ZFMISC_1:31;
A28: Sgm Q2j is one-to-one by FINSEQ_3:92;
A29: Q2j c= Seg width A by A26,A27,XBOOLE_1:8;
A30: rng Sgm Q2j=Q2j by FINSEQ_1:def 14;
     {i0} c= dom A by A7,A8,ZFMISC_1:31;
    then P2i c= dom A by A23,XBOOLE_1:8;
    then
A33: [:P2i,Q2j:] c= Indices A by A29,ZFMISC_1:96;
A34: dom Sgm P2i=Seg card P2i by FINSEQ_3:40;
    i in {i} by TARSKI:def 1;
    then
A35: i in P2i by XBOOLE_0:def 3;
A36: not i in P2 by A2,A16,XBOOLE_0:def 5;
    then
A37: card P2i=card P2+1 by CARD_2:41;
    then
A38: card P2i-'1=card P9 by A18,NAT_D:34;
A39: not j in Q2 by A3,A17,XBOOLE_0:def 5;
    then
A40: card Q2j=card Q2+1 by CARD_2:41;
    then
A41: ESS = SS by A13,A18,A19,A36,CARD_2:41,MATRIX13:def 3;
    j in {j} by TARSKI:def 1;
    then j in Q2j by XBOOLE_0:def 3;
    then consider y being object such that
A42: y in dom Sgm Q2j and
A43: Sgm Q2j.y=j by A30,FUNCT_1:def 3;
    rng Sgm P2i=P2i by FINSEQ_1:def 14;
    then consider x being object such that
A44: x in dom Sgm P2i and
A45: Sgm P2i.x=i by A35,FUNCT_1:def 3;
    reconsider x,y as Element of NAT by A44,A42;
    -1_K<>0.K by VECTSP_1:28;
    then
A46: power(K).(-1_K,x+y)<>0.K by Lm2;
    set L=LaplaceExpL(ESS,x);
A47: dom L = Seg len L by FINSEQ_1:def 3
      .= Seg card P2i by LAPLACE:def 7;
    then
A48: y in dom L by A13,A18,A19,A37,A40,A42,FINSEQ_3:40;
A49: dom Sgm Q2j=Seg card Q2j by FINSEQ_3:40;
    then Delete(ESS,x,y) = EqSegm(A,P2i\{i},Q2j\{j}) by A13,A18,A19,A37,A40,A34
,A44,A45,A42,A43,MATRIX13:64
      .= EqSegm(A,P2,Q2j\{j}) by A36,ZFMISC_1:117
      .= EqSegm(A,P2,Q2) by A39,ZFMISC_1:117
      .= Segm(A,P2,Q2) by A13,A18,A19,MATRIX13:def 3
      .= EqSegm(S,P9,Q9) by A13,A20,MATRIX13:def 3;
    then
A50: power(K).(-1_K,x+y) * Det Delete(ESS,x,y)<>0.K by A15,A38,A46,VECTSP_1:12;
A51: Indices ESS = [:Seg card P2i,Seg card P2i:] by MATRIX_0:24;
    then
A52: [x,y] in Indices ESS by A13,A18,A19,A37,A40,A34,A49,A44,A42,ZFMISC_1:87;
A53: rng Sgm R=R by FINSEQ_1:def 14;
    now
      let k such that
A54:  k in dom L and
A55:  k <> y;
      Sgm Q2j.k <> j by A13,A18,A19,A37,A40,A49,A28,A42,A43,A47,A54,A55,
FUNCT_1:def 4;
      then
A56:  not Sgm Q2j.k in {j} by TARSKI:def 1;
      Sgm Q2j.k in Q2j by A13,A18,A19,A37,A40,A30,A49,A47,A54,FUNCT_1:def 3;
      then Sgm Q2j.k in Q2 by A56,XBOOLE_0:def 3;
      then
A57:  Sgm Q2j.k in Q by A17;
      then
A58:  Sgm Q2j.k in R by A5;
      consider z being object such that
A59:  z in dom Sgm R and
A60:  Sgm R.z=Sgm Q2j.k by A5,A53,A57,FUNCT_1:def 3;
      reconsider z as Element of NAT by A59;
      [x,k] in Indices ESS by A34,A44,A47,A51,A54,ZFMISC_1:87;
      then ESS*(x,k) = A*(i,Sgm Q2j.k) by A45,A41,MATRIX13:def 1
        .= Line(A,i).(Sgm R.z) by A24,A60,A58,MATRIX_0:def 7
        .= (card R |-> 0.K).z by A6,A59,FUNCT_1:13
        .= 0.K by A25,A59,FINSEQ_2:57;
      hence L.k = 0.K*Cofactor(ESS,x,k) by A54,LAPLACE:def 7
        .= 0.K;
    end;
    then
A61: L.y = Sum L by A48,MATRIX_3:12
      .= Det ESS by A34,A44,LAPLACE:25;
    L.y = SS*(x,y)*Cofactor(ESS,x,y) by A48,A41,LAPLACE:def 7
      .= A*(i,j)*(power(K).(-1_K,x+y)*Det Delete(ESS,x,y)) by A45,A43,A41,A52,
MATRIX13:def 1;
    then Det ESS<>0.K by A4,A61,A50,VECTSP_1:12;
    then the_rank_of A >= card P2i by A13,A18,A19,A37,A40,A33,MATRIX13:def 4;
    hence thesis by A14,A18,A37,NAT_1:13;
  end;
end;
