theorem Th27:
  for tr be Element of Permutations(n+2) st tr is
  being_transposition holds tr is odd
proof
  set K = the Fanoian Field;
  let tr be Element of Permutations(n+2);
  assume tr is being_transposition;
  then sgn(tr,K)=-1_K by Th14;
  hence thesis by Th23;
end;
