reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th46:
  for M be (Matrix of n,K),perm,Perm st perm = Perm holds Det(M*
  Perm) = -(Det(M),perm)
proof
  let M be (Matrix of n,K),perm,Perm such that
A1: Perm=perm;
  per cases;
  suppose
A2: n<2;
    then perm=idseq n by Lm3;
    then
A3: M*perm=M by Th39;
    perm is even by A2,Lm3;
    hence thesis by A1,A3,MATRIX_1:def 16;
  end;
  suppose
    n>=2;
    then reconsider n2=n-2 as Nat by NAT_1:21;
    reconsider M9=M as Matrix of n2+2,K;
    reconsider Perm2=Perm as Permutation of Seg(n2+2);
    reconsider perm2=perm as Element of Permutations(n2+2);
    Det(M9*Perm2) = sgn(perm2,K)*Det(M9) by A1,Th45;
    hence thesis by Th26;
  end;
end;
