The string of inequalities in Eq. (2.15) is not quite a proof, because technically there is no such thing as \\( v + w + u \\), for example.

Instead, there is \\( (v+w)+u \\) and \\( v+(w+u) \\), and so on.

1. Formally prove, using only the rules of symmetric monoidal preorders ( [Definition 2.2](https://forum.azimuthproject.org/discussion/1974) ), that given the assertions in Eq. (2.13), the conclusion in Eq. (2.14) follows.

2. Reflexivity and transitivity should show up in your proof. Make sure you are explicit about where they do.

3. How can you look at the wiring diagram Eq. (2.6) and know that the commutative axiom (Definition 2.1 d.) does not need to be invoked?

**Eq 2.15** \\( t + u \le v + w + u \le v + x + z \le y + z \\)

**Eq 2.14** \\( t + u \le y + z \\)

**Eq 2.13** \\(t \le v + w \\) and \\( w + u \le x + z \\) and \\( v + x \le y \\)

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Instead, there is \\( (v+w)+u \\) and \\( v+(w+u) \\), and so on.

1. Formally prove, using only the rules of symmetric monoidal preorders ( [Definition 2.2](https://forum.azimuthproject.org/discussion/1974) ), that given the assertions in Eq. (2.13), the conclusion in Eq. (2.14) follows.

2. Reflexivity and transitivity should show up in your proof. Make sure you are explicit about where they do.

3. How can you look at the wiring diagram Eq. (2.6) and know that the commutative axiom (Definition 2.1 d.) does not need to be invoked?

**Eq 2.15** \\( t + u \le v + w + u \le v + x + z \le y + z \\)

**Eq 2.14** \\( t + u \le y + z \\)

**Eq 2.13** \\(t \le v + w \\) and \\( w + u \le x + z \\) and \\( v + x \le y \\)

[Previous](https://forum.azimuthproject.org/discussion/1974)

[Next](https://forum.azimuthproject.org/discussion/1976)