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;

theorem Th26:
  i in Seg n & j in Seg n & nt.i = nt.j & i<>j implies Det Segm(M,
  nt,nt1) = 0.K
proof
  assume that
A1: i in Seg n and
A2: j in Seg n and
A3: nt.i = nt.j and
A4: i<>j;
A5: i<j or j<i by A4,XXREAL_0:1;
  Line(Segm(M,nt,nt1),i) = Line(Segm(M,nt,nt1),j) by A1,A2,A3,Th25;
  hence thesis by A1,A2,A5,MATRIX11:50;
end;
