reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;

theorem
  for K being Field, n being Element of NAT st n >= 1 holds Per (0.(K,n,
  n)) = 0.K
proof
  let K be Field, n be Element of NAT;
  set B = In (Permutations n, Fin Permutations n);
  set f = PPath_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: 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
A2: 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 A2,FINSUB_1:7,FUNCOP_1:7;
  end;
  assume
A3: n>=1;
A4: for x being object st x in dom PPath_product(0.(K,n,n)) holds (
  PPath_product(0.(K,n,n))).x = (Permutations(n) --> 0.K).x
  proof
    let x be object;
    assume x in dom PPath_product(0.(K,n,n));
    for p being Element of Permutations n holds (Permutations(n) --> 0.K).
    p = (the multF of K) $$ Path_matrix(p,(0.(K,n,n)))
    proof
      defpred P[Nat] means (the multF of K) $$ (($1+1) |-> 0.K)=0.K;
      let p be Element of Permutations(n);
A5:   for k being Nat st P[k] holds P[k+1]
      proof
        let k be Nat;
A6:     (k+1+1) |-> 0.K = ((k+1) |-> 0.K) ^ <* 0.K *> by FINSEQ_2:60;
        assume P[k];
        then (the multF of K) $$ ( (k+1+1) |-> 0.K) = (0.K)*(0.K) by A6,
FINSOP_1:4
          .= 0.K;
        hence thesis;
      end;
A7:   for i,j st i in dom (n |-> 0.K) & j=p.i holds (n |-> 0.K).i = (0.(K,
      n,n))*(i,j)
      proof
        let i,j;
        assume that
A8:     i in dom (n |-> 0.K) and
A9:     j=p.i;
A10:    i in Seg n by A8,FUNCOP_1:13;
        then j in Seg n by A9,MATRIX_7:14;
        then
A11:    j in Seg width (0.(K,n,n)) by A3,MATRIX_0:23;
        i in dom (0.(K,n,n)) by A10,MATRIX_7:13;
        then [i,j] in [:dom (0.(K,n,n)),Seg width (0.(K,n,n)):] by A11,
ZFMISC_1:def 2;
        then
A12:    [i,j] in Indices (0.(K,n,n)) by MATRIX_0:def 4;
        (n |-> 0.K).i=0.K by A10,FUNCOP_1:7;
        hence thesis by A12,MATRIX_3:1;
      end;
      len (n |-> 0.K) = n by CARD_1:def 7;
      then
A13:  Path_matrix(p,(0.(K,n,n))) = n |-> 0.K by A7,MATRIX_3:def 7;
A14:  n-'1+1=n by A3,XREAL_1:235;
      1 |-> 0.K = <* 0.K *> by FINSEQ_2:59;
      then
A15:  P[0] by FINSOP_1:11;
      for k being Nat holds P[k] from NAT_1:sch 2(A15,A5);
      then (the multF of K) $$ Path_matrix(p,(0.(K,n,n))) = 0.K by A13,A14;
      hence thesis by FUNCOP_1:7;
    end;
    hence thesis by Def1;
  end;
  dom (Permutations(n) --> 0.K) = Permutations(n) by FUNCOP_1:13;
  then dom (PPath_product(0.(K,n,n))) = dom (Permutations(n) --> 0.K) by
FUNCT_2:def 1;
  then
A16: PPath_product(0.(K,n,n)) = Permutations(n) --> 0.K by A4,FUNCT_1:2;
A17: 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 A16,FUNCOP_1:7;
  end;
A18: 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;
A19:  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 A16,FUNCOP_1:7,FVSUM_1:6;
      hence thesis by A19,BINOP_1:3;
    end;
  end;
  Permutations n in Fin Permutations n by FINSUB_1:def 5; then
  In (Permutations n, Fin Permutations n) = Permutations n
    by SUBSET_1:def 8; then
  In (Permutations n, Fin Permutations n) <> {} & G0.B=0.K by FUNCOP_1:7;
  hence thesis by A1,A17,A18,SETWISEO:def 3;
end;
