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The Mission of the Third Person of the Most Blessed Trinity is, in fact, inseparable from the mission of Peter and of his successors for the visible unity and unified government of the Church. Communion with the faithful means above all communion with the visible Peter, assurance of communion with the invisible Spirit of God.
Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. ...
And therefore, if a man refuse to hear the Church, let him be considered - so the Lord commands - as a heathen and a publican. It follows that those who are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit. ...
Let every one then abhor sin, which defiles the mystical members of our Redeemer; but if anyone unhappily falls and his obstinacy has not made him unworthy of communion with the faithful, let him be received with great love, and let eager charity see in him a weak member of Jesus Christ. ...
Now since its Founder willed this social body of Christ to be visible, the cooperation of all its members must also be externally manifest through their profession of the same faith and their sharing the same sacred rites, through participation in the same Sacrifice, and the practical observance of the same laws. Above all, it is absolutely necessary that the Supreme Head, that is, the Vicar of Jesus Christ on earth, be visible to the eyes of all, since it is He who gives effective direction to the work which all do in common in a mutually helpful way towards the attainment of the proposed end. As the Divine Redeemer sent the Paraclete, the Spirit of Truth, who in His name should govern the Church in an invisible way, so, in the same manner, He commissioned Peter and his successors to be His personal representatives on earth and to assume the visible government of the Christian community.
Pius XII, Mystici Corporis Christi
May the primacy of Peter, entrusted to poor human beings, be always exercised in its original sense as desired by the Lord, so that it may be recognized still more in its true meaning by our brothers as yet not in full communion with us.
Benedict XVI