reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th40:
  Permutations 0 = {<*>NAT}
proof
  now
    let p be object;
    assume p in Permutations(0);
    then reconsider q=p as Permutation of Seg 0 by MATRIX_1:def 12;
    q=<*>NAT;
    hence p in {<*>NAT} by TARSKI:def 1;
  end;
  then
A1: Permutations(0) c={<*>NAT};
  {<*>NAT} c=Permutations(0)
  proof
    let x be object;
    assume x in {<*>NAT};
    then x is Permutation of Seg 0 by Lm4,TARSKI:def 1;
    hence thesis by MATRIX_1:def 12;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
