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You are telling me that the human spirit is actually a component of our flesh,"
Spirit can be defined as breath and breath has oxygen which makes us alive, keeps us alive.
So no, in that sense, the breath/spirit is effectively oxygen in us, you know the us = a living soul lol.
So the breath/spirit in us is not another identity, so to speak, independent of the flesh = living soul.
What this living soul produces or can produced, (in another sense) is or can be referred to as the soul or spirit of us (the package) and these products arrive from or are driven from the brain, the memory bank.
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therefore contradicting the verse "that which is flesh is flesh....."And to keep that flesh alive, it needs the breath/spirit of life (breath also being defined as spirit).
So it is Not a component of the flesh, but what the flesh draws in.
You need to understand the context of the passage that you are drawing from.
The passage that you are quoting, is speaking of and showing us, how you are or can be born twice, 1st from your mother = flesh and 2nd born of the Spirit of God and that 2nd birth is Not the 1st, to which we are covering.
John 3:6 - That which is born of the flesh is flesh; and that which is born of the Spirit is spirit.
More clarification or another context -> John 6:63 - It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you (Fathers, who is Spirit), they are spirit, and they are life.
The flesh aspect that profits nothing here, is the words of it, as words are the Key topic here.
Hence the words Jesus was speaking, were from the Father who is spirit, hence his words are deemed a spirit.
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The living soul of Kam, produces the spirit, soul, heart, mind of the Kam, in all his ways and living etc. 
