theorem Th15:
  idseq n= 1_Group_of_Perm(n)
proof
  reconsider e=idseq n as Element of Group_of_Perm(n) by Th11;
  reconsider e9=idseq n as Element of Permutations(n) by Def12;
  for p being Element of Group_of_Perm(n) holds p * e=p & e* p=p
  proof
    let p be Element of Group_of_Perm(n);
    reconsider p9=p as Element of Permutations(n) by Def13;
A1: e * p =p9 * e9 by Def13
      .=p by Th12;
    p * e =e9 * p9 by Def13
      .=p by Th12;
    hence thesis by A1;
  end;
  hence thesis by GROUP_1:4;
end;
