reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);

theorem Th11:
  sgn(p2,K) = 1_K or sgn(p2,K) = -1_K
proof
  set KK=the carrier of K;
  set n2=n+2;
  set 2S=2Set Seg n2;
  set mm=the multF of K;
  set Path=Part_sgn(p2,K);
  2S in Fin 2S by FINSUB_1:def 5; then
  In(2S,Fin 2S)=2S by SUBSET_1:def 8;
  then reconsider 2S9=2S as Element of Fin 2S;
  consider G be Function of Fin 2S, KK such that
A2: mm $$(2S9,Path) = G.2S9 and
A3: for e be Element of KK st e is_a_unity_wrt mm holds G.{} = e and
A4: for s holds G.{s} = Path.s and
A5: for B be Element of Fin 2S st B c= 2S9 & B <> {} for s st s in 2S9 \
  B holds G.(B \/ {s}) = mm.(G.B,Path.s) by SETWISEO:def 3;
  defpred P[Nat] means for B be Element of Fin 2S st card B=$1 & B c= 2S holds
  (G.B=1_K or G.B=-1_K);
A6: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A7: P[k];
    set k1=k+1;
    let B be Element of Fin 2S such that
A8: card B=k1 and
A9: B c= 2S;
    now
      per cases;
      case
        k=0;
        then consider x being object such that
A10:    B={x} by A8,CARD_2:42;
        x in B by A10,TARSKI:def 1;
        then reconsider x as Element of 2S by A9;
        G.B=Path.x by A4,A10;
        hence thesis by Th5;
      end;
      case
A11:    k>0;
        consider x being object such that
A12:    x in B by A8,CARD_1:27,XBOOLE_0:def 1;
        reconsider x as Element of 2S by A9,A12;
        B\{x} c= 2S by A9;
        then reconsider B9=B\{x} as Element of Fin 2S by FINSUB_1:def 5;
A13:    not x in B9 by ZFMISC_1:56;
A14:    {x}\/ B9=B by A12,ZFMISC_1:116;
        then
A15:    k+1=card B9+1 by A8,A13,CARD_2:41;
        then
A16:    G.B9=1_K or G.B9=-1_K by A7,A9,XBOOLE_1:1;
        x in 2S\B9 by A13,XBOOLE_0:def 5;
        then G.B=mm.(G.B9,Path.x) by A5,A9,A11,A14,A15,CARD_1:27,XBOOLE_1:1;
        then G.B=1_K*1_K or G.B=1_K*(-1_K) or G.B=(-1_K)*1_K or G.B=(-1_K)*(-
        1_K) by A16,Th5;
        then G.B=1_K*1_K or G.B=1_K*(-1_K) by VECTSP_1:10;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A17: P[0]
  proof
    let B be Element of Fin 2S such that
A18: card B=0 and
    B c= 2S;
    B={} by A18;
    hence thesis by A3,FVSUM_1:4;
  end;
  for k be Nat holds P[k] from NAT_1:sch 2(A17,A6);
  then P[card 2S9];
  hence thesis by A2;
end;
