reserve k,n,i,j for Nat;

theorem Th35:
  for n being Nat, K being commutative Ring, p being Element of
Permutations(n), A being Matrix of n,K st n>=1 holds (Path_product(A@)).(p") =
  (Path_product(A)).p
proof
  let n be Nat, K be commutative Ring,
    p be Element of Permutations n, A be
  Matrix of n,K;
  assume
A1: n>=1;
A2: len (Path_matrix(p,A))=n by MATRIX_3:def 7;
  then reconsider g=Path_matrix(p,A)*(p") as FinSequence of K by A1,Th33;
  (Path_product(A@)).(p") = -((the multF of K) $$ Path_matrix(p",A@),p")
  by MATRIX_3:def 8
    .= -((the multF of K) $$ g,p") by A1,Th34
    .= -((the multF of K) $$ g,p) by A1,Th28
    .= -((the multF of K) $$ (Path_matrix(p,A)),p) by A2,Th30,Th31;
  hence thesis by MATRIX_3:def 8;
end;
