1 Corinthians 11:11-12

Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. For as the woman is of the man, even so is the man also by the woman; but all things of God (KJV)

Then: Paul is moving on in his reasons for distinguishing the sexes in a worship service. He turns from Creation's evidence to the interrelation of men and women. Following the Lord's initial action, the existence of all succeeding generations comes from those which preceded. No man nor woman exsits independently of the other. They serve in unity and community.

Now: The obvious meaning relates to the ongoing reproduction of mankind from generation to generation. Paul calls upon that to support his teaching that men and women should worship together in a seemly, ordered fashion. In short, order in creation requires order in worship. To signify both order and headship, some rely upon an extended meaning of women wearing head coverings. Phillips (p. 241) understands them to be the means by which the woman can assume a man's role of teaching, preaching, reading scripture, praying, or singing - any of the leadership functions - when wearing a head cloth because the cloth signfies her being under authority. That point of view assumes roles in the kingdom are based on sex, and not on individual gifts and talents bestowed by the Spirit. It sidesteps Paul's earlier presentation and does not explain the perception of limited acceptance into the Lord's kingdom. Phillips (p. 239) makes an unsupported assertion that women are not inferior to men. Within all top-to-bottom analyses of herarchies something is always superior to something else. That holds true for an authoritarian church. There is no other logical conclusion. This conclusion conflicts, however, with Paul's later comparison of the church as a single, integral unit (the church or a human body) which operates using reciprocal submission.

Extra source: Exploring 1 Corinthians: an expository commentary by John Phillips. Grand Rapids MI: Kregel Publications, c2002.