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);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th42:
  for p2,q2 st q2 = p2" holds sgn(p2,K) = sgn(q2,K)
proof
A1: n+1+1>=0+1 by XREAL_1:6;
  let p2,q2;
  assume q2=p2";
  then
A2: -(1_K,p2)=-(1_K,q2) by A1,MATRIX_7:29;
A3: -(1_K,q2)=sgn(q2,K)*1_K by Th26;
  -(1_K,p2)=sgn(p2,K)*1_K by Th26;
  then sgn(p2,K)*1_K=sgn(q2,K) by A2,A3,VECTSP_1:def 4;
  hence thesis;
end;
