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
  for M,P1,Q,Q1 st card P1 = card Q1 & Q c= Q1 & Det EqSegm(M,P1,Q1) <>
  0.K ex P st P c= P1 & card P = card Q & Det EqSegm(M,P,Q) <> 0.K
proof
  let M,P1,Q,Q1 such that
A1: card P1 = card Q1 and
A2: Q c= Q1 and
A3: Det EqSegm(M,P1,Q1) <> 0.K;
  defpred R[Nat] means for Q st Q c= Q1 & card Q=$1 ex P st P c= P1 & card P=
  card Q & Det EqSegm(M,P,Q) <> 0.K;
A4: for k be Element of NAT st k < card Q1 & R[k+1] holds R[k]
  proof
    let k be Element of NAT such that
A5: k < card Q1 and
A6: R[k+1];
    let Q such that
A7: Q c= Q1 and
A8: card Q=k;
    Q c< Q1 by A5,A7,A8,XBOOLE_0:def 8;
    then Q1\Q<>{} by XBOOLE_1:105;
    then consider x being object such that
A9: x in Q1\Q by XBOOLE_0:def 1;
    reconsider x as non zero Element of NAT by A9;
    reconsider Qx=Q\/{x} as without_zero finite Subset of NAT;
A10: not x in Q by A9,XBOOLE_0:def 5;
    then
A11: card Qx=k+1 by A8,CARD_2:41;
    x in Q1 by A9,XBOOLE_0:def 5;
    then {x} c= Q1 by ZFMISC_1:31;
    then Qx c= Q1 by A7,XBOOLE_1:8;
    then consider P2 such that
A12: P2 c= P1 and
A13: card Qx=card P2 and
A14: Det EqSegm(M,P2,Qx) <> 0.K by A6,A8,A10,CARD_2:41;
A15: k+1-'1=k+1-1 by XREAL_0:def 2;
    x in {x} by TARSKI:def 1;
    then
A16: x in Qx by XBOOLE_0:def 3;
A18: dom Sgm Qx = Seg card Qx by FINSEQ_3:40;
    rng Sgm Qx = Qx by FINSEQ_1:def 14;
    then consider j be object such that
A19: j in Seg card Qx and
A20: Sgm Qx.j=x by A18,A16,FUNCT_1:def 3;
    set E=EqSegm(M,P2,Qx);
    reconsider j as Element of NAT by A19;
    consider i such that
A21: i in Seg card Qx and
A22: Det Delete(E,i,j) <> 0.K by A13,A14,A19,Lm4;
    take P=P2\{Sgm P2.i};
A23: P c= P2 by XBOOLE_1:36;
A24: Qx\{x}=Q by A10,ZFMISC_1:117;
    then card P=card Q by A13,A19,A20,A21,Th64;
    hence thesis by A8,A11,A12,A13,A19,A20,A21,A22,A24,A23,A15,Th64;
  end;
A25: R[card Q1]
  proof
    let Q;
    assume that
A26: Q c= Q1 and
A27: card Q=card Q1;
    Q=Q1 by A26,A27,CARD_2:102;
    hence thesis by A1,A3;
  end;
  for k be Element of NAT st k <= card Q1 holds R[k]
   from PRE_POLY:sch 1(A25,A4);
  hence thesis by A2,NAT_1:43;
end;
