reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th51:
  for M being Matrix of n, K st (ex i being Element of NAT st i in
Seg n & for k being Element of NAT st k in Seg n holds Col(M,i).k = 0.K) holds
  (the addF of K) $$ (In(Permutations n,Fin Permutations n), Path_product M) 
  = 0.K
proof
  let M be Matrix of n, K;
  reconsider n1=n as Element of NAT by ORDINAL1:def 12;
  reconsider M1=M as Matrix of n1, K;
  given i being Element of NAT such that
A1: i in Seg n & for k being Element of NAT st k in Seg n holds Col(M,i)
  .k = 0.K;
  set P1 = In(Permutations n,Fin Permutations n);
  set f = Path_product M1;
  set F = the addF of K;
  Permutations n in Fin Permutations n by FINSUB_1:def 5; then
  P1 = Permutations n by SUBSET_1:def 8; then
  reconsider P1 as non empty Element of Fin Permutations n1;
  defpred P[non empty Element of Fin Permutations n1] means F $$ ($1, f) = 0.K;
A2: for x being Element of Permutations n1, B being non empty Element of Fin
  Permutations n1 st x in P1 & B c= P1 & not x in B & P[B] holds P[B \/ {.x.}]
  proof
    let x be Element of Permutations n1, B be non empty Element of Fin
    Permutations n1;
    assume that
    x in P1 and
    B c= P1 and
A3: not x in B and
A4: P[B];
    F $$ (B \/ {.x.}, f) = F.(F $$(B,f), f.x) by A3,SETWOP_2:2
      .= F $$(B,f) + 0.K by A1,Th50
      .= 0.K by A4,RLVECT_1:4;
    hence thesis;
  end;
A5: for x being Element of Permutations n1 st x in P1 holds P[{.x.}]
  proof
    let x be Element of Permutations n1;
    assume x in P1;
    F $$ ({.x.}, f) = f.x by SETWISEO:17
      .= 0.K by A1,Th50;
    hence thesis;
  end;
  P[P1] from NonEmptyFiniteX(A5,A2);
  hence thesis;

end;
