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
  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 Det M
  = 0.K
proof
  let M be Matrix of n,K;
  assume
A1: 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;
  set f = Path_product M;
  set F = the addF of K;
  Det M = F $$ (In(Permutations n,Fin Permutations n),f) by MATRIX_3:def 9
    .= 0.K by A1,Th51;
  hence thesis;
end;
