Should we wear a covering in church? 1 Corinthians 11

1 Corinthians 11:3 (NASBStr)

But I want you to understand that Christ is the head of every man, and the man is the head of a woman, and God is the head of Christ.

Should a man wear a head covering while he is praying or prophesying?   Should a women be required to wear a head covering during worship, or praying or prophesying during a church service?  These are the questions that Paul addresses in the first part of chapter eleven.   Last year I made a point on this discussion of how trivial a matter like this should be to the entire church doctrine.   Paul seems to be diverted from the major theme of Christ resurrection by handling such a topic.   Yet, when we see the above verse that Paul is not arguing for men to be uncovered and women to be covered from a social or community tradition point of view.   (NOTE:  whatever  you argue is the interpretation of this argument Paul has explicitly stated it is in regard to praying or prophesying in a church service.)  In the above verse Paul states the first phase of his argument as pertaining to the nature of the Trinity and the nature of Creation.   You and I and the church may not like the practical implications of head covering in the church but Paul is not giving us an oral argument based upon his "customs" or "customs of Corinth" at the time.   Paul is telling us that covering in the church is based upon the fact that by "function" (not nature") Christ is subjective to God; and the women by "function" (not by "nature") is subjective to man.   His argument seems to be that since this is true, than in the church service, the women should be wearing some sign of the subjection when we she is praying or prophesying (NOTE: the fact that he makes this argument means women can't be "silent" in the church all the time ... 1 Corinthians 14:34.)   It seems plain that Paul is telling us that the covering of the head is a picture of our submissiveness to Christ and to each other.   Women, by nature, are on the same level as men.  Christianity did that.  However, by function in the church (and the home) to be submissive to the man.  The man, is to be submissive to the women since she is part of the Body of Christ and we are to submit ourselves to each other.   The "covering" in the church is a symbol of that submission.  If we refuse, for practical reasons, to wear a real covering (should we refuse?) than we ought to at least make sure we follow the spirit of chapter eleven, if not the letter of Paul's instruction.  It is based upon the function of the trinity and the order of creation ... not the whim of an unmarried male-teacher named Paul.