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);

theorem Th14:
  for tr be Element of Permutations(n+2) st tr is
  being_transposition holds sgn(tr,K) = -1_K
proof
  set n2=n+2;
  set S=Seg n2;
  let tr be Element of Permutations(n2) such that
A1: tr is being_transposition;
  reconsider Tr=tr as Permutation of S by MATRIX_1:def 12;
  reconsider Id=idseq n2,IdTr=(id S)*Tr as Element of Permutations(n2) by
MATRIX_1:def 12;
  rng Tr=S by FUNCT_2:def 3;
  then IdTr=Tr by RELAT_1:54;
  then sgn(tr,K) = -sgn(Id,K) by A1,Th13;
  hence thesis by Th12;
end;
