“It shall be lawful for each and all to withdraw without penalty and at any time from obedience and loyalty, even if he has been enthroned, and to avoid him as a magician, a heathen, a publican and a heresiarch.”

We hold that Bergoglio is an antipope due to his election being invalid because the See was not vacant. The conclave itself was invalidly called, since Pope Benedict never lawfully resigned. This is the true base premise, according to the available evidence. However, the heretic Bergoglio was also ineligible to be elected, even if it had been a valid conclave, by reason of his manifest public heresy placing him outside the Church. A non-Catholic cannot be pope. So either way, Bergoglio is not and never has been pope.

Thirdly, if somehow he had become pope, and then taught his myriad heresies, we would have a sacred duty to withdraw from him, not tolerate him out of respect to the Office (which he does not hold). This third possibility would seem remote, since it is Catholic Dogma that Christ Himself* bestows the Office (not the Cardinals nor the Church) and had He done so in the case of Bergoglio, would have nullified His promise in Luke 22:32. The whole point of the Papacy is the errorless Rock. Here are some true popes and various saints talking about this third scenario:

Catholic Church Teaching on Heresy and the Pope:

• In addition, [by this Our Constitution, which is to remain valid in perpetuity We enact, determine, decree and define] that if ever at any time it shall appear that any Bishop, even if he be acting as an Archbishop, Patriarch or Primate; or any Cardinal of the aforesaid Roman Church, or, as has already been mentioned, any legate, or even the Roman Pontiff, prior to his promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic Faith or fallen into some heresy:
  (i) the promotion or elevation, even if it shall have been uncontested and by the unanimous assent of all the Cardinals, shall be null, void and worthless… “Cum ex Apostolatus Officio” Apostolic Constitution of Pope Paul IV, 1559
• It shall be lawful for each and all of the cardinals,…as well as for all the clergy and the Roman people,… to withdraw without penalty and at any time from obedience and loyalty to the person so elected even if he has been enthroned (while they themselves, notwithstanding this, remain fully committed to the faith of the Roman church and to obedience towards a future Roman pontiff entering office in accordance with the canons) and to avoid him as a magician, a heathen, a publican and a heresiarch.”
  Pope Julius II, Council of Lateran V. 1513
• “Now when [the Pope] is explicitly a heretic, he falls ipso facto from his dignity and out of the Church, and the Church must either deprive him, or, as some say, declare him deprived, of his Apostolic See.”
  St. Francis de Sales, “The Catholic Controversy”
• “…a pope who is a manifest heretic by that fact ceases to be pope and head, just as he by that fact ceases to be a Christian and a member of the body of the Church; wherefore he can be judged and punished by the Church. This is the judgment of all the early fathers, who teach that manifest heretics immediately lose all jurisdiction.”
  St. Robert Bellarmine, “On the Roman Pontiff”
• “If God permitted a pope to be notoriously heretical and contumacious, he would then cease to be pope, and the Apostolic Chair would be vacant.”
  St. Alphonsus de Liguori, “The Truths of the Faith”
• “An heretical pope necessarily ceases to be head of the Church, for by his heresy he is no longer a member thereof: in the event of his still claiming the Roman see a general council, improperly so-called because without the pope, could remove him. But this is not deposition, since by his own act he is no longer pope.”
  A Catholic Dictionary, 1951. Deposition
• “The councils of Constance and Basle, and Gallican theologians, hold that a council may depose a pope…(2) /ob fidem/ (on account of his faith or rather want of faith, i.e. heresy). In point of fact however, heresy is the only legitimate ground. For a heretical pope has ceased to be a member of the Church, and cannot, therefore, be its head.”
  Catholic Encyclopedia, 1913. [Vol. IV p.435] Councils

Prophesies of Heresy from Scripture:

• Take heed to yourselves, and to the whole flock, wherein the Holy Ghost hath placed you bishops, to rule the church of God, which he hath purchased with his own blood. I know that, after my departure, ravening wolves will enter in among you, not sparing the flock. And of your own selves shall arise men speaking perverse things, to draw away disciples after them. Acts 20:28-30
• A man that is a heretic, after the first and second admonition, avoid: Knowing that he, that is such an one, is subverted, and sinneth, being condemned by his own judgment. Titus 3:10-11
• Whosoever revolteth, and continueth not in the doctrine of Christ, hath not God. He that continueth in the doctrine, the same hath both the Father and the Son. If any man come to you, and bring not this doctrine, receive him not into the house nor say to him, God speed you. For he that saith unto him, God speed you, communicateth with his wicked works. 2 John 1:9-11
• Now the Spirit manifestly saith, that in the last times some shall depart from the faith, giving heed to spirits of error, and doctrines of devils, Speaking lies in hypocrisy, and having their conscience seared….” 1 Tim 4:1-2
• For there shall be a time, when they will not endure sound doctrine; but, according to their own desires, they will heap to themselves teachers, having itching ears: And will indeed turn away their hearing from the truth, but will be turned unto fables. 2 Tim 4:3-4
• But there were also false prophets among the people, even as there shall be among you lying teachers, who shall bring in sects of perdition, and deny the Lord who bought them: bringing upon themselves swift destruction. And many shall follow their riotousnesses, through whom the way of truth shall be evil spoken of. And through covetousness shall they with feigned words make merchandise of you. Whose judgment now of a long time lingereth not, and their perdition slumbereth not. 2 Peter 2:1-3

http://www.catholicessentials.net/heresy.htm

*PA Ch2 p3. Therefore whoever succeeds to the chair of Peter obtains by the institution of Christ himself, the primacy of Peter over the whole Church. “So what the truth has ordained stands firm, and blessed Peter perseveres in the rock-like strength he was granted, and does not abandon that guidance of the Church which he once received” [47]. HERE