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 Th27:
  for tr be Element of Permutations(n+2) st tr is
  being_transposition holds tr is odd
proof
  set K = the Fanoian Field;
  let tr be Element of Permutations(n+2);
  assume tr is being_transposition;
  then sgn(tr,K)=-1_K by Th14;
  hence thesis by Th23;
end;
