Be true to thyself, meaning to be true to God who is our real self (Our Father). God used the word thyself, because Thy (capital T) is used in ecclesiastical language to address God (i.e. Thy will be done). The use of the small letter t signifies being the child of God.
So, to give up our life to God means to be and to behave like a true child of God, this is who and what man is.
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