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 Th35:
  for perm be Element of Permutations n st nt1 = nt2 * perm holds
  Det Segm(M,nt1,nt) = -(Det Segm(M,nt2,nt),perm) & Det Segm(M,nt,nt1) = -(Det
  Segm(M,nt,nt2),perm)
proof
  let perm be Element of Permutations n such that
A1: nt1=nt2*perm;
  reconsider Perm=perm as Permutation of Seg n by MATRIX_1:def 12;
  Segm(M,nt1,nt) = Segm(M,nt2,nt) * Perm by A1,Th33;
  hence Det Segm(M,nt1,nt) = -(Det Segm(M,nt2,nt),perm) by MATRIX11:46;
  thus Det Segm(M,nt,nt1) = Det Segm(M,nt,nt1)@ by MATRIXR2:43
    .= Det (Segm(M,nt,nt2)@ * Perm) by A1,Th34
    .= -(Det (Segm(M,nt,nt2)@),perm) by MATRIX11:46
    .= -(Det Segm(M,nt,nt2),perm) by MATRIXR2:43;
end;
