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

theorem ThFunc1:
  for f being Function holds
    f is ext-natural-valued iff for x being object holds f.x is ext-natural
proof
  let f be Function;
  hereby
    assume A1: f is ext-natural-valued;
    let x be object;
    per cases;
    suppose x in dom f;
      hence f.x is ext-natural by A1;
    end;
    suppose not x in dom f;
      hence f.x is ext-natural by FUNCT_1:def 2;
    end;
  end;
  thus (for x being object holds f.x is ext-natural)
    implies f is ext-natural-valued;
end;
