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 Th9:
  for p2,q2,pq2,i,j st pq2 = p2*q2 & q2 is being_transposition & q2
.i = j & i < j for s st Part_sgn(p2,K).s <> Part_sgn(pq2,K).s holds i in s or j
  in s
proof
  set n2=n+2;
  let p, q, pq be Element of Permutations(n2), i, j such that
A1: pq = p*q and
A2: q is being_transposition and
A3: q.i=j and
A4: i < j;
  reconsider q9=q,pq9=pq as Permutation of Seg n2 by MATRIX_1:def 12;
  let s be Element of 2Set Seg(n2) such that
A5: Part_sgn(p,K).s<>Part_sgn(pq,K).s;
A6: dom q9=Seg n2 by FUNCT_2:52;
A7: dom pq9=Seg n2 by FUNCT_2:52;
  assume that
A8: not i in s and
A9: not j in s;
  consider i9,j9 be Nat such that
A10: i9 in Seg n2 and
A11: j9 in Seg n2 and
A12: i9 < j9 and
A13: s={i9,j9} by Th1;
A14: j<>j9 by A13,A9,TARSKI:def 2;
A15: j<>i9 by A13,A9,TARSKI:def 2;
  i<>j9 by A13,A8,TARSKI:def 2;
  then q.j9=j9 by A2,A3,A4,A11,A14,A6,Th8;
  then
A16: pq.j9=p.j9 by A1,A11,A7,FUNCT_1:12;
  i<>i9 by A13,A8,TARSKI:def 2;
  then q.i9=i9 by A2,A3,A4,A10,A15,A6,Th8;
  then pq.i9=p.i9 by A1,A10,A7,FUNCT_1:12;
  hence contradiction by A5,A10,A11,A12,A13,A16,Th6;
end;
