reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem
  for F being set st(for X being set st X in F holds X is ext-natural-membered)
  holds union F is ext-natural-membered
proof
  let F be set;
  assume A1: for X being set st X in F holds X is ext-natural-membered;
  let x be object;
  assume x in union F;
  then consider Y being set such that
    A2: x in Y & Y in F by TARSKI:def 4;
  Y is ext-natural-membered by A1, A2;
  hence x is ext-natural by A2;
end;
