Complete the proof of **Proposition 2.35** by proving that the three remaining conditions of [Definition 2.2](https://forum.azimuthproject.org/discussion/1976) are satisfied.

**Proposition 2.35**.
Suppose \$$\mathcal{X} = (X, \le) \$$ is a preorder and \$$\mathcal{X}^{op} = ( X, \ge ) \$$ is its opposite.
If \$$(X, \le, I, \otimes ) \$$ is a symmetric monoidal preorder then so is its opposite, \$$(X, \gt, I, \otimes ) \$$ .

(i) monoidal unit : an element \$$I \in X \$$

(ii) monoidal product : a function \$$\otimes : X \times X \rightarrow X \$$

These constituents must satisfy the following properties:
**Proof** of 2.1.a

Suppose \$$x_1 \ge y_1 \$$ and \$$x_2 \ge y_2 \$$ in \$$\mathcal{X}^{op} \$$ ;
we need to show that \$$x_1 \otimes x_2 \ge y_1 \otimes y_2 \$$.
But by definition of opposite order, we have \$$y_1 \le x_1 \$$
and \$$y_2 \le x_2 \$$ in \$$\mathcal{X} \$$, and thus
\$$y_1 \otimes y_2 \le x_1 \otimes x_2 \$$ in \$$\mathcal{X} \$$.
Thus indeed \$$x_1 \otimes x_2 \ge y_1 \otimes y_2 \$$ in \$$\mathcal{X}^{op} \$$ .

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