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 Th50:
  for p being Element of Permutations n 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 (Path_product M).p = 0.K
proof
  let p be Element of Permutations n;
  let M be Matrix of n, K;
A1: (Path_product M).p = -((the multF of K) $$ Path_matrix(p,M),p) by
MATRIX_3:def 8
    .= - (Product (Path_matrix(p,M)),p) by GROUP_4:def 2;
  assume 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;
  then consider l being Element of NAT such that
A2: l in Seg n and
A3: Path_matrix (p,M).l = 0.K by Th49;
  set k = l;
  len Path_matrix (p, M) = n by MATRIX_3:def 7;
  then k in dom (Path_matrix(p,M)) by A2,FINSEQ_1:def 3;
  then
A4: Product Path_matrix(p,M) = 0.K by A3,FVSUM_1:82;
  per cases;
  suppose
    p is even;
    hence thesis by A4,A1,MATRIX_1:def 16;
  end;
  suppose
    p is odd;
    then -(Product (Path_matrix(p,M)),p) = -Product (Path_matrix(p,M)) by
MATRIX_1:def 16
      .= 0.K by A4,RLVECT_1:12;
    hence thesis by A1;
  end;
end;
