Theorem1.5.1. If ,then.


Proof. Assume  . Then x, 2-x and 2+x are all nonnegative.

Therefore, the product of these terms is also nonnegative. Adding 1 to this product gives a positive number, so:

Multiplying out on the left side proves that

as claimed.                                                                                                             􏰐

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also, you could write your answer as comment.

New Concept

Proof Method


In order to prove that  IMPLIES :

1. Write,“Assume.”

2. Show that  logically follows.

Logical Deductions

Logical deductions, or inference rules, are used to prove new propositions using previously proved ones.

A fundamental inference rule is modus ponens. This rule says that a proof of P together with a proof that P IMPLIES Q is a proof of Q.

Inference rules are sometimes written in a funny notation. For example, modus ponens is written:


When the statements above the line, called the antecedents, are proved, then we can consider the statement below the line, called the conclusion or consequent, to also be proved.

A key requirement of an inference rule is that it must be sound: an assignment of truth values to the letters P, Q, ..., that makes all the antecedents true must also make the consequent true. So if we start off with true axioms and apply sound inference rules, everything we prove will also be true.

There are many other natural, sound inference rules, for example:



On the other hand,


is not sound: if P is assigned T and Q is assigned F, then the antecedent is true and the consequent is not.

As with axioms, we will not be too formal about the set of legal inference rules. Each step in a proof should be clear and “logical”; in particular, you should state what previously proved facts are used to derive each new conclusion.