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 Th43:
  for M be (Matrix of n+2,K),perm2,Perm2 st perm2=Perm2 for p2,q2
  st q2 = p2*Perm2" holds (Path_product(M)).q2 = sgn(perm2,K)*(Path_product(M*
  Perm2)).p2
proof
  let M be (Matrix of n+2,K),perm2,Perm2 such that
A1: perm2=Perm2;
  set P=Permutations(n+2);
  set mm=the multF of K;
  let p2,q2 such that
A2: q2=p2*Perm2";
  reconsider perm29=perm2" as Element of P by MATRIX_7:18;
  set PM=mm $$ (Path_matrix(q2,M));
  set PMp=mm $$ (Path_matrix(p2,M*Perm2));
  sgn(q2,K)=sgn(p2,K)*sgn(perm29,K) by A1,A2,Th24
    .=sgn(p2,K)*sgn(perm2,K) by Th42;
  then -(PM,q2)=(sgn(perm2,K)*sgn(p2,K))*PM by Th26
    .=sgn(perm2,K)*(sgn(p2,K)*PM) by GROUP_1:def 3
    .=sgn(perm2,K)*(sgn(p2,K)*PMp) by A2,Th41
    .=sgn(perm2,K)*-(PMp,p2) by Th26
    .=sgn(perm2,K)*((Path_product(M*Perm2)).p2) by MATRIX_3:def 8;
  hence thesis by MATRIX_3:def 8;
end;
