reserve k,n,i,j for Nat;

theorem
  for K being Ring, n being Nat st n>=1 holds Det (0.(K,n,n)
  ) = 0.K
proof
  let K be Ring, n be Nat;
  set B=In(Permutations n,Fin Permutations n);
  set f=Path_product(0.(K,n,n));
  set F=the addF of K;
  set Y=the carrier of K;
  set X=Permutations n;
  reconsider G0= Fin X --> 0.K as Function of Fin X, Y;
A1: G0.B=0.K by FUNCOP_1:7;
A2: for e being Element of Y st e is_a_unity_wrt F holds G0.({}) = e
  proof
    let e be Element of Y;
    0.K is_a_unity_wrt F by FVSUM_1:6;
    then
A3: F.(0.K,e)=e by BINOP_1:3;
    assume e is_a_unity_wrt F;
    then F.(0.K,e)=0.K by BINOP_1:3;
    hence thesis by A3,FINSUB_1:7,FUNCOP_1:7;
  end;
  assume
A4: n>=1;
A5: for x being object st x in dom (Path_product(0.(K,n,n))) holds (
  Path_product(0.(K,n,n))).x=(Permutations(n) --> 0.K).x
  proof
    let x be object;
    assume x in dom (Path_product(0.(K,n,n)));
    for p being Element of Permutations(n) holds (Permutations(n) --> 0.K)
    .p = -((the multF of K) $$ Path_matrix(p,(0.(K,n,n))),p)
    proof
      defpred P[Nat] means (the multF of K) $$ ( ($1+1) |-> 0.K)=0.
      K;
      let p be Element of Permutations(n);
A6:   for k being Nat st P[k] holds P[k+1]
      proof
        let k be Nat;
A7:     (k+1+1) |-> 0.K = ((k+1) |-> 0.K) ^ <* 0.K *> by FINSEQ_2:60;
        assume P[k];
        then (the multF of K) $$ ( (k+1+1) |-> 0.K) = (0.K)*(0.K) by A7,
FINSOP_1:4
          .= 0.K;
        hence thesis;
      end;
      (1 |-> 0.K)= <* 0.K *> by FINSEQ_2:59;
      then
A8:   P[0] by FINSOP_1:11;
A9:   for k being Nat holds P[k] from NAT_1:sch 2(A8,A6);
A10:  now
        per cases;
        case
          p is even;
          hence -(0.K,p)=(0.K) by MATRIX_1:def 16;
        end;
        case
          not p is even;
          then -(0.K,p)= -(0.K) by MATRIX_1:def 16
            .=0.K by RLVECT_1:12;
          hence -(0.K,p)=(0.K);
        end;
      end;
A11:  for i,j st i in dom (n |-> 0.K) & j=p.i holds (n |-> 0.K).i=(0.(K,n,
      n))*(i,j)
      proof
        let i,j;
        assume that
A12:    i in dom (n |-> 0.K) and
A13:    j=p.i;
A14:    i in Seg n by A12,FUNCOP_1:13;
        then j in Seg n by A13,Th14;
        then
A15:    j in Seg width (0.(K,n,n)) by A4,MATRIX_0:23;
        i in dom (0.(K,n,n)) by A14,Th13;
        then
A16:    [i,j] in Indices (0.(K,n,n)) by A15,ZFMISC_1:def 2;
        (n |-> 0.K).i=0.K by A14,FUNCOP_1:7;
        hence thesis by A16,MATRIX_3:1;
      end;
      len (n |-> 0.K)=n by CARD_1:def 7;
      then
A17:  Path_matrix(p,(0.(K,n,n)))=(n |-> 0.K) by A11,MATRIX_3:def 7;
      n-'1=n-1 by A4,XREAL_1:233;
      then
A18:  n-'1+1=n;
      (Permutations(n) --> 0.K).p = 0.K by FUNCOP_1:7;
      hence thesis by A17,A9,A18,A10;
    end;
    hence thesis by MATRIX_3:def 8;
  end;
  dom (Permutations(n) --> 0.K)=Permutations(n) by FUNCOP_1:13;
  then dom (Path_product(0.(K,n,n)))=dom (Permutations(n) --> 0.K) by
FUNCT_2:def 1;
  then
A19: Path_product(0.(K,n,n))=Permutations(n) --> 0.K by A5,FUNCT_1:2;
A20: for x being Element of X holds G0.({x}) = f.x
  proof
    let x be Element of X;
    G0.({.x.})=0.K by FUNCOP_1:7;
    hence thesis by A19,FUNCOP_1:7;
  end;
A21: for B9 being Element of Fin X st B9 c= B & B9 <> {} for x being Element
  of X st x in B\B9 holds G0.(B9 \/ {x}) = F.(G0.B9,f.x)
  proof
    let B9 be Element of Fin X;
    assume that
    B9 c= B and
    B9 <> {};
    thus for x being Element of X st x in B\B9 holds G0.(B9 \/ {x}) = F.(G0.B9
    ,f.x)
    proof
      let x be Element of X;
      assume x in B\B9;
A22:  G0.(B9 \/ {.x.})=0.K & G0.B9=0.K by FUNCOP_1:7;
      f.x= 0.K & 0.K is_a_unity_wrt F by A19,FUNCOP_1:7,FVSUM_1:6;
      hence thesis by A22,BINOP_1:3;
    end;
  end;
  X in Fin X by FINSUB_1:def 5; then
  B = X by SUBSET_1:def 8; then
  B <> {} or F is having_a_unity;
  then
  (the addF of K) $$ (In(Permutations(n),Fin Permutations n),
Path_product(0.(K,n,n)))=
  0.K by A1,A2,A20,A21,SETWISEO:def 3;
  hence thesis by MATRIX_3:def 9;
end;
