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The proofs of the above Lemmas are given in Appendix: Proofs.
The proofs of and are obvious.
The proofs of (11), (12) are similar.
The proofs of others are similar.
The proofs of God's existence are not conclusive, but neither are the proofs of God's non-existence.
The proofs of the others are similar.
The proofs of Corollaries 1.1 and 1.2 are similar.
The proofs of the other cases are similar.
The proofs of these lemmas are shown in Appendix.
The proofs of the above lemmas are almost obvious.
The proofs of the main results are provided in Sects.
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