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## Homework Statement

Prologue Chapter 1. Problem 8.

(P10) Trichotomy law: For every number a, one and only one of the following holds:

(i) a = 0

(ii) a is in the collection P

(iii) -a is in the collection P

(P11) If a and b are in P, then a + b are in P

(P12) If a and b are in P, then a x b is in P.

## The Attempt at a Solution

P10: Let (b-a) be any number d. Then P'10 reduces to

i) d = 0

ii) 0 <d

iii) 0 >d

This is in essence what P10 is, except with the variable being different

P11: Let there be numbers a, b, c, d.

If a<b and c<d, then by P'12 the following are true:

1) a+c < b+c

2) b+c < b+d

By P'11, a + c < b + d

Looking at the case where 0 < b and 0 < d, b + d > 0 which is P11.

P12: If 0 < a and 0 < c, by P'13, 0 < ac, which is P12