theorem Th14:
  for tr be Element of Permutations(n+2) st tr is
  being_transposition holds sgn(tr,K) = -1_K
proof
  set n2=n+2;
  set S=Seg n2;
  let tr be Element of Permutations(n2) such that
A1: tr is being_transposition;
  reconsider Tr=tr as Permutation of S by MATRIX_1:def 12;
  reconsider Id=idseq n2,IdTr=(id S)*Tr as Element of Permutations(n2) by
MATRIX_1:def 12;
  rng Tr=S by FUNCT_2:def 3;
  then IdTr=Tr by RELAT_1:54;
  then sgn(tr,K) = -sgn(Id,K) by A1,Th13;
  hence thesis by Th12;
end;
