Well-Defined Relation/Examples/Less Than on Congruence Modulo 6

Example of Well-Defined Relation
Let $x \mathrel {C_6} y$ be the equivalence relation defined on the natural numbers as congruence modulo $6$:


 * $x \mathrel {C_6} y \iff x \equiv y \pmod 6$

defined as:


 * $\forall x, y \in \N: x \equiv y \pmod 6 \iff \exists k, l, m \in \N, m < 6: 6 k + m = x \text { and } 6 l + m = y$

Let $\eqclass x {C_6}$ denote the equivalence class of $x$ under $C_6$.

Let $\N / {C_6}$ denote the quotient set of $\N$ by $C_6$.

Let us define the relation $L$ on $\N / {C_6}$ as follows:


 * $\tuple {\eqclass x {C_6}, \eqclass y {C_6} } \in L \iff \exists k, l, m, n \in \N, m < n < 6: x = 6 k + m, y = 6 l + n$

Then $L$ is a well-defined relation.

Proof
Let:
 * $x \mathrel {C_6} x'$
 * $y \mathrel {C_6} y'$

for arbitrary $x, y, x', y' \in \N$.

We need to demonstrate that:
 * $\tuple {\eqclass x {C_6}, \eqclass y {C_6} } \in L \iff \tuple {\eqclass {x'} {C_6}, \eqclass {y'} {C_6} } \in L$

So: