The Cartesian product can be used in Graphing equivalence, a general property in Mathematics, especially in Arithmetic.

The relation of eqivalence has the properties:

- reflexivity
- symmetry
- transitivity

On the other hand, the relation of total order has the properties:

- irreflexivity
- asymmetry
- transitivity
- totalness

The Cartesian graph of equivalence can be transformed into one for total order by striking out the terms which allow for reflexivity and symmetry.

Given a "square product", reflexivity is graphed by the **diagonal** of the Cartesian product, wherein the ordered pair has the same first and second members. If the diagonal is struck out, irreflexivity results.

Again, given a "square product", its diagonal separates the "square" into an **upper triangular form** and a **lower triangular form** such that upper form mirrors lower form. Thus, if (a,b) appears in upper form, then (b,a) appears in the lower form; and vice versa. Then striking out one of these forms, say, the lower triangular form, achieves asymmetry.

It will then be found that the totalness property also exists: **every term is either first member of an ordered pair of the form, or second member, but not both**.

Satisfaction of the properties of total order by such a product means that it graphs the property of total order.

Result is that the **upper subtriangle** of a "square" Cartesian product **graphs total order**. (The **lower subtriangle** would do just as well.)

( | (a,b) | (a,c) | (a,d) | (a,e) |

( | ( | (b,c) | (b,d) | (b,e) |

( | ( | () | (c,d) | (c,e) |

( | ( | ( | ( | (d,e) |

( | ( | ( | ( | ( |

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