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Prove the following, please state the theorems used: If a < b and c < d, then a + c < b + d. If

Prove the following, please state the theorems used:

If a < b and c < d, then a + c < b + d.

If a ≤ b and c ≤ d, then a + c ≤ b + d.

 If a>b and b>c, then a>c.

If a > 0 and b < 0, then ab < 0.

If a < b and c > 0, then ac < bc.

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