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 Th68:
  for P,Q,i,j0 st j0 in Seg n9 & i in P & not j0 in P & card P =
card Q & [:P,Q:] c= Indices M9 holds card P = card (P\{i}\/{j0}) & [:P\{i}\/{j0
},Q:] c= Indices M9 & (Det EqSegm(RLine(M9,i,Line(M9,j0)),P,Q) = Det EqSegm(M9,
P\{i}\/{j0},Q) or Det EqSegm(RLine(M9,i,Line(M9,j0)),P,Q) = -Det EqSegm(M9,P\{i
  }\/{j0},Q))
proof
  let P,Q,i,j0 such that
A1: j0 in Seg n9 and
A2: i in P and
A3: not j0 in P and
A4: card P = card Q and
A5: [:P,Q:] c= Indices M9;
  set Pi=P\{i};
A6: Pi c= P by XBOOLE_1:36;
  set SQ=Sgm Q;
  set Pij=Pi\/{j0};
  set SPij=Sgm Pij;
A7: rng SPij=Pij by FINSEQ_1:def 14;
  card P>0 by A2;
  then reconsider C=card P - 1 as Element of NAT by NAT_1:20;
  card P=C+1;
  then
A8: card Pi=C by A2,STIRL2_1:55;
  not j0 in Pi by A3,XBOOLE_0:def 5;
  hence
A9: card Pij = C+1 by A8,CARD_2:41
    .= card P;
  then reconsider SPij,SQ9=SQ as Element of (card P)-tuples_on NAT by A4;
A10: Segm(M9,SPij,SQ9) = Segm(M9,Pij,Q) by A4,A9
    .= EqSegm(M9,Pij,Q) by A4,A9,Def3;
  P c= Seg len M9 by A4,A5,Th67;
  then
A11: Pi c= Seg len M9 by A6;
  n9=len M9 by MATRIX_0:def 2;
  then {j0} c= Seg len M9 by A1,ZFMISC_1:31;
  then
A12: Pij c= Seg len M9 by A11,XBOOLE_1:8;
  set SP=Sgm P;
A13: rng SQ=Q by FINSEQ_1:def 14;
  rng SP=P by FINSEQ_1:def 14;
  then consider PT be Element of (card P)-tuples_on NAT such that
A14: rng PT=Pij and
A15: Segm(RLine(M9,i,Line(M9,j0)),SP,SQ) = Segm(M9,PT,SQ) by A2,A5,A13,Th39;
  Q c= Seg width M9 by A4,A5,Th67;
  hence [:Pij,Q:] c= Indices M9 by A4,A9,A12,Th67;
  EqSegm(RLine(M9,i,Line(M9,j0)),P,Q) = Segm(RLine(M9,i,Line(M9,j0)),P,Q)
  by A4,Def3
    .= Segm(M9,PT,SQ9) by A4,A15;
  hence thesis by A9,A7,A13,A14,A10,Th36;
end;
