reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th41:
  for K being Ring
  for A being Matrix of 0,K holds Det A = 1.K
proof
  let K be Ring;
  reconsider kk=idseq 0 as Element of Permutations(0) by Th40,TARSKI:def 1;
  let A be Matrix of 0,K;
A1: (Path_product A).(idseq 0)=1.K
  proof
    reconsider p = idseq 0 as Element of Permutations(0)
      by Lm4,MATRIX_1:def 12;
A2: -(1_K,p)=1_K
    proof
      reconsider q = id Seg 0 as Element of Permutations(0) by Lm4,
MATRIX_1:def 12;
      len Permutations 0 = 0 by MATRIX_1:9;
      then q is even by MATRIX_1:16;
      hence thesis by MATRIX_1:def 16;
    end;
    1_K is_a_unity_wrt (the multF of K) by GROUP_1:21;
    then len Path_matrix(p,A) = 0 & (the multF of K) is having_a_unity by
MATRIX_3:def 7,SETWISEO:def 2;
    then
    (the multF of K) $$ Path_matrix(p,A) = the_unity_wrt (the multF of K)
    by FINSOP_1:def 1
      .= 1_K by GROUP_1:22;
    hence thesis by A2,MATRIX_3:def 8;
  end;
  Permutations 0 in Fin Permutations 0 by FINSUB_1:def 5; then
A3: In(Permutations(0),Fin Permutations 0) = Permutations(0)
  by SUBSET_1:def 8
    .= {. kk .} by Th40;
  Det A=(the addF of K) $$ (In(Permutations(0),Fin Permutations 0),
Path_product(A)) by
MATRIX_3:def 9
    .=1.K by A1,A3,SETWISEO:17;
  hence thesis;
end;
