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 Th9:
  len Permutations(n)=n
proof
  set x = the Element of Permutations(n);
  reconsider q=x as Permutation of Seg n by Def12;
A1: dom q=Seg n by FUNCT_2:52;
  then reconsider q as FinSequence by FINSEQ_1:def 2;
  n in NAT by ORDINAL1:def 12;
  then len q=n by A1,FINSEQ_1:def 3;
  hence thesis by Def11;
end;
