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 Th26:
  -(a,perm2) = sgn(perm2,K)*a
proof
  A1: len (Permutations (n + 2)) = n + 2 by MATRIX_1:9;
  per cases;
  suppose
A2: perm2 is even;
    then sgn (perm2,K) = 1_K by A1,Th15;
    hence sgn (perm2,K)*a = a
                         .= -(a,perm2) by A2,MATRIX_1:def 16;
  end;
  suppose A10: perm2 is odd;
    consider P being FinSequence of Group_of_Perm (n + 2) such that
    A11: perm2 = Product P & for i st i in dom P
      ex trans be Element of Permutations(n+2) st P.i = trans &
    trans is being_transposition by Th21;
    for l being FinSequence of (Group_of_Perm (n + 2))
    st perm2 = Product l
     & (for i being Nat st i in dom l
        ex q being Element of Permutations (n + 2)
        st (l.i = q & q is being_transposition))
    holds (len l) mod 2 = 1 by A1, A10, NAT_D:12;
    then A13: (len P) mod 2 = 1 by A11;
    A14: -(a,perm2) = -a by A10,MATRIX_1:def 16;
    sgn(perm2,K) = -1_K by A13,A11,Th15;
    hence sgn(perm2,K)*a = (-1_K) * a
                        .= -a by VECTSP_2:29
                        .= -(a,perm2) by A14;
  end;
end;
