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 Th52:
  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 (PPath_product M).p = 0.K
proof
  let p be Element of Permutations n;
  let M be Matrix of n, K;
  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
A1: l in Seg n and
A2: Path_matrix (p,M).l = 0.K by Th49;
  len Path_matrix (p, M) = n by MATRIX_3:def 7;
  then l in dom Path_matrix(p,M) by A1,FINSEQ_1:def 3;
  then
A3: Product Path_matrix(p,M) = 0.K by A2,FVSUM_1:82;
  (PPath_product M).p = (the multF of K) $$ Path_matrix(p,M) by Def1
    .= 0.K by A3,GROUP_4:def 2;
  hence thesis;
end;
