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;

theorem
  ex P being permutational non empty set st len P =n
proof
  set p = the Permutation of Seg n;
  set P={p};
  now
    take n;
    let x;
    assume x in P;
    hence x is Permutation of Seg n by TARSKI:def 1;
  end;
  then reconsider P as permutational non empty set by Def10;
  take P;
  len P= n
  proof
    set x = the Element of P;
    reconsider y=x as Function of Seg n,Seg n by TARSKI:def 1;
A1: dom y=Seg n by FUNCT_2:52;
    then reconsider s=y as FinSequence by FINSEQ_1:def 2;
    n in NAT & len P= len s by Def11,ORDINAL1:def 12;
    hence thesis by A1,FINSEQ_1:def 3;
  end;
  hence thesis;
end;
