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 Th5:
  Part_sgn(p2,K).s = 1_K or Part_sgn(p2,K).s = -1_K
proof
  consider i,j such that
A1: i in Seg (n+2) and
A2: j in Seg (n+2) and
A3: i < j and
A4: s={i,j} by Th1;
  p2 is Permutation of Seg (n+2) by MATRIX_1:def 12;
  then p2.i <> p2.j by A1,A2,A3,FUNCT_2:19;
  then p2.i > p2.j or p2.i < p2.j by XXREAL_0:1;
  hence thesis by A1,A2,A3,A4,Def1;
end;
