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 Th6:
  for i,j st i in Seg (n+2) & j in Seg (n+2) & i < j & p2.i = q2.i
  & p2.j = q2.j holds Part_sgn(p2,K).{i,j} = Part_sgn(q2,K).{i,j}
proof
  set n2=n+2;
  let i,j such that
A1: i in Seg n2 and
A2: j in Seg n2 and
A3: i<j and
A4: p2.i=q2.i and
A5: p2.j=q2.j;
  reconsider p29=p2 as Permutation of Seg n2 by MATRIX_1:def 12;
A6: p29.i<>p29.j by A1,A2,A3,FUNCT_2:19;
  now
    per cases by A6,XXREAL_0:1;
    suppose
A7:   p2.i < p2.j;
      then Part_sgn(p2,K).{i,j}=1_K by A1,A2,A3,Def1;
      hence thesis by A1,A2,A3,A4,A5,A7,Def1;
    end;
    suppose
A8:   p2.i > p2.j;
      then Part_sgn(p2,K).{i,j}=-1_K by A1,A2,A3,Def1;
      hence thesis by A1,A2,A3,A4,A5,A8,Def1;
    end;
  end;
  hence thesis;
end;
