reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  K for non empty doubleLoopStr,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D,
  F for add-associative right_zeroed
  right_complementable Abelian non empty doubleLoopStr;
reserve A,B for Matrix of n,K;
reserve A,A9,B,B9,C for Matrix of n,F;
reserve i,j,n for Nat,
  K for Field,
  a,b for Element of K;
reserve x,y,x1,x2,y1,y2 for set,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Field,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of K;
reserve p,q for Element of Permutations(n);

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;
