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 Th10:
  for p2,q2,pq2,i,j,K st pq2 = p2*q2 & q2 is being_transposition &
q2.i = j & i < j & 1_K <> -1_K holds Part_sgn(p2,K).{i,j} <> Part_sgn(pq2,K).{i
  ,j} & for k st k in Seg(n+2) & i <> k & j <> k holds Part_sgn(p2,K).{i,k} <>
  Part_sgn(pq2,K).{i,k} iff Part_sgn(p2,K).{j,k} <> Part_sgn(pq2,K).{j,k}
proof
  set n2=n+2;
  let p, q, pq be Element of Permutations(n2), i, j, K such that
A1: pq = p*q and
A2: q is being_transposition and
A3: q.i=j and
A4: i < j and
A5: 1_K <> -1_K;
A6: i in dom q by A2,A3,A4,Th8;
  set P2=Part_sgn(pq,K);
  set P1=Part_sgn(p,K);
  reconsider p9=p,q9=q,pq9=pq as Permutation of Seg n2 by MATRIX_1:def 12;
A7: dom q9=Seg n2 by FUNCT_2:52;
A8: j in dom q by A2,A3,A4,Th8;
A9: dom pq9=Seg n2 by FUNCT_2:52;
  then
A10: pq.i=p.j by A1,A3,A6,A7,FUNCT_1:12;
  q.j=i by A2,A3,A4,Th8;
  then
A11: pq.j=p.i by A1,A8,A9,A7,FUNCT_1:12;
  dom p9=Seg n2 by FUNCT_2:52;
  then
A12: p9.i<>p9.j by A4,A6,A8,A7,FUNCT_1:def 4;
  now
    per cases by A12,XXREAL_0:1;
    suppose
A13:  p.i < p.j;
      then P1.{i,j}=1_K by A4,A6,A8,A7,Def1;
      hence P1.{i,j}<>P2.{i,j} by A4,A5,A6,A8,A7,A10,A11,A13,Def1;
    end;
    suppose
A14:  p.i > p.j;
      then P1.{i,j}=-1_K by A4,A6,A8,A7,Def1;
      hence P1.{i,j}<>P2.{i,j} by A4,A5,A6,A8,A7,A10,A11,A14,Def1;
    end;
  end;
  hence P1.{i,j}<>P2.{i,j};
  let k such that
A15: k in Seg n2 and
A16: i <> k and
A17: j <> k;
A18: q.k=k by A2,A3,A4,A7,A15,A16,A17,Th8;
A19: pq.k=p.(q.k) by A1,A9,A15,FUNCT_1:12;
  i<k or k<i by A16,XXREAL_0:1;
  then
A20: {i,k} in 2Set Seg n2 by A6,A7,A15,Th1;
A21: P1.{i,k} = P2.{i,k} implies P1.{j,k} = P2.{j,k}
  proof
A22: j<k or k<j by A17,XXREAL_0:1;
A23: i<k or i>k by A16,XXREAL_0:1;
    assume
A24: P1.{i,k} = P2.{i,k};
    P1.{k,i}=1_K or P1.{k,i}=-1_K by A20,Th5;
    then
    pq.j<pq.k & p.j<p.k or pq.j>pq.k & p.j>p.k by A5,A6,A7,A10,A11,A15,A18,A19
,A24,A23,Lm1;
    then P2.{j,k}=1_K & P1.{j,k}=1_K or P2.{j,k}=-1_K & P1.{j,k}=-1_K by A8,A7
,A15,A22,Def1;
    hence thesis;
  end;
  j<k or k<j by A17,XXREAL_0:1;
  then
A25: {j,k} in 2Set Seg n2 by A8,A7,A15,Th1;
  P1.{j,k} = P2.{j,k} implies P1.{i,k} = P2.{i,k}
  proof
A26: i<k or k<i by A16,XXREAL_0:1;
A27: j<k or j>k by A17,XXREAL_0:1;
    assume
A28: P1.{j,k} = P2.{j,k};
    P1.{k,j}=1_K or P1.{k,j}=-1_K by A25,Th5;
    then
    pq.i<pq.k & p.i<p.k or pq.i>pq.k & p.i>p.k by A5,A8,A7,A10,A11,A15,A18,A19
,A28,A27,Lm1;
    then P2.{i,k}=1_K & P1.{i,k}=1_K or P2.{i,k}=-1_K & P1.{i,k}=-1_K by A6,A7
,A15,A26,Def1;
    hence thesis;
  end;
  hence thesis by A21;
end;
