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 Th23:
  K is Fanoian non degenerated implies
  ( perm2 is even iff sgn(perm2,K) = 1_K )&
  ( perm2 is odd iff sgn(perm2,K) = -1_K )
proof
  assume A0: K is Fanoian non degenerated;
  set n2=n+2;
A1: len Permutations n2 = n2 by MATRIX_1:9;
  thus
A2: perm2 is even implies sgn(perm2,K) = 1_K by A1,Th15;
  thus sgn(perm2,K)= 1_K implies perm2 is even
  proof
    assume
A3: sgn(perm2,K)= 1_K;
    consider P be FinSequence of Group_of_Perm(n2) such that
A4: perm2=Product P and
A5: for i st i in dom P ex trans be Element of Permutations(n2) st P.i
    =trans & trans is being_transposition by Th21;
    assume perm2 is odd;
    then len P mod 2 <> 0 by A1,A4,A5;
    then len P mod 2 = 1 by NAT_D:12;
    then sgn(perm2,K)=-1_K by A4,A5,Th15;
    hence thesis by A0,A3,th22;
  end;
  hence thesis by A0,A2,Th11,th22;
end;
