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 Th31:
  not mt is one-to-one implies Det Segm(M,mt1,mt) = 0.K
proof
  assume not mt is one-to-one;
  then consider x,y being object such that
A1: x in dom mt and
A2: y in dom mt and
A3: mt.x=mt.y and
A4: x<>y;
A5: dom mt=Seg m by FINSEQ_2:124;
  then consider i be Nat such that
A6: x=i and
A7: 1<=i and
A8: i<=m by A1;
  consider j be Nat such that
A9: y=j and
A10: 1<=j and
A11: j<=m by A2,A5;
A12: j in Seg m by A10,A11;
  i in Seg m by A7,A8;
  hence thesis by A3,A4,A6,A9,A12,Th30;
end;
