reserve k,n,i,j for Nat;

theorem Th8:
  for a being Element of Group_of_Perm 2 st (ex q being Element of
  Permutations 2 st q=a & q is being_transposition) holds a = <*2,1*>
proof
  let a be Element of Group_of_Perm 2;
  given q being Element of Permutations 2 such that
A1: q=a and
A2: q is being_transposition;
  the carrier of Group_of_Perm 2 = Permutations 2 by MATRIX_1:def 13;
  then reconsider b=a as Permutation of Seg 2 by MATRIX_1:def 12;
  ex i,j being Nat st i in dom q & j in dom q & i<>j & q. i=j & q.j=i & for
  k being Nat st k <>i & k<>j & k in dom q holds q.k=k by A2;
  then b is being_transposition by A1;
  hence thesis by Th4;
end;
