Problem 8

Define properties of inequality in terms of $<$ instead of $P$. From (P'10) if $b = 0$ then:

• (i) $a = 0$
• (ii) $a < 0 \rightarrow -a > 0 \rightarrow -a \in P$
• (iii) $a > 0 \rightarrow a \in P$

Except for the order of statementst that is (P10).

(P11) is derived like the following: \[ \begin{align*} b &> 0 \\ b + a &> 0 + a && \text{| (P'12)} \\ b + a &> a && \text{| with (P'11) and } a > 0 \\ b + a &> 0 && \text{| and that is (P11)} \\ \end{align*} \]

(P12) is derived like this: \[ \begin{align*} a &> 0 \\ ab &> 0b \\ ab &> 0 && \text{| as long as } b > 0 \\ \end{align*} \]

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