Theorem: Multiplication of Cross-Relation Equivalence Classes on Natural Numbers is Well-Defined

Let $\struct {\N, +}$ be the semigroup of natural numbers under addition.
Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.
Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:
:$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
Let $\eqclass {x, y} {}$ denote the equivalence class of $\tuple {x, y}$ under $\boxtimes$.
Let $\otimes$ be the binary operation defined on these equivalence classes as:
:$\forall \eqclass {a, b} {}, \eqclass {c, d} {} \in \N \times \N: \eqclass {a, b} {} \otimes \eqclass {c, d} {} = \eqclass {\tuple {a \cdot c} + \tuple {b \cdot d}, \tuple {a \cdot d} + \tuple {b \cdot c} } {}$
where $a \cdot c$ denotes natural number multiplication between $a$ and $c$.
{{WIP|Introduce the language of the Definition:Quotient Set.}}
The operation $\otimes$ on these equivalence classes is well-defined, in the sense that:
{{begin-eqn}}
{{eqn | l = \eqclass {a_1, b_1} {}
      | r = \eqclass {a_2, b_2} {}
      | c = 
}}
{{eqn | l = \eqclass {c_1, d_1} {}
      | r = \eqclass {c_2, d_2} {}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \eqclass {a_1, b_1} {} \otimes \eqclass {c_1, d_1} {}
      | r = \eqclass {a_2, b_2} {} \otimes \eqclass {c_2, d_2} {}
      | c = 
}}
{{end-eqn}}

Proof:
Let $\eqclass {a_1, b_1} {}, \eqclass {a_2, b_2} {}, \eqclass {c_1, d_1} {}, \eqclass {c_2, d_2} {}$ be $\boxtimes$-equivalence classes such that $\eqclass {a_1, b_1} {} = \eqclass {a_2, b_2} {}$ and $\eqclass {c_1, d_1} {} = \eqclass {c_2, d_2} {}$.
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a_1, b_1} {}
      | r = \eqclass {a_2, b_2} {}
      | c = {{Defof|Operation Induced by Direct Product}}
}}
{{eqn | lo= \land
      | l = \eqclass {c_1, d_1} {}
      | r = \eqclass {c_2, d_2} {}
      | c = {{Defof|Operation Induced by Direct Product}}
}}
{{eqn | ll= \leadstoandfrom
      | l = a_1 + b_2
      | r = a_2 + b_1
      | c = {{Defof|Cross-Relation}}
}}
{{eqn | lo= \land
      | l = c_1 + d_2
      | r = c_2 + d_1
      | c = {{Defof|Cross-Relation}}
}}
{{end-eqn}}
Both $+$ and $\times$ are commutative and associative on $\N$. Thus:
{{begin-eqn}}
{{eqn | l = \paren {a_1 \cdot c_1 + b_1 \cdot d_1 + a_2 \cdot d_2 + b_2 \cdot c_2}
      | o = +
      | r = \paren {a_2 \cdot c_1 + b_2 \cdot c_1 + a_2 \cdot d_1 + b_2 \cdot d_1}
      | c = 
}}
{{eqn | lo= =
      | l = \paren {a_1 + b_2} \cdot c_1 + \paren {b_1 + a_2} \cdot d_1
      | o = +
      | r = a_2 \cdot \paren {c_1 + d_2} + b_2 \cdot \paren {d_1 + c_2}
      | c = 
}}
{{eqn | lo= =
      | l = \paren {b_1 + a_2} \cdot c_1 + \paren {a_1 + b_2} \cdot d_1
      | o = +
      | r = a_2 \cdot \paren {d_1 + c_2} + b_2 \cdot \paren {c_1 + d_2}
      | c = as $a_1 + b_2 = b_1 + a_2, c_1 + d_2 = d_1 + c_2$
}}
{{eqn | lo= =
      | l = b_1 \cdot c_1 + a_2 \cdot c_1 + a_1 \cdot b_2 + a_1 \cdot d_1
      | o = +
      | r = a_2 \cdot d_1 + a_2 \cdot c_2 + b_2 \cdot c_1 + b_2 \cdot d_2
      | c = 
}}
{{eqn | lo= =
      | l = \paren {a_1 \cdot d_1 + b_1 \cdot c_1 + a_2 \cdot c_2 + b_2 \cdot d_2}
      | o = +
      | r = \paren {a_2 \cdot c_1 + b_2 \cdot c_1 + a_2 \cdot d_1 + b_2 \cdot d_1}
      | c = 
}}
{{end-eqn}}
So we have $a_1 \cdot c_1 + b_1 \cdot d_1 + a_2 \cdot d_2 + b_2 \cdot c_2 = a_1 \cdot d_1 + b_1 \cdot c_1 + a_2 \cdot c_2 + b_2 \cdot d_2$ and so, by the definition of $\boxtimes$, we have:
{{explain|The above uses cancellability of natural numbers under addition, which has been glossed over so far and needs to be made explicit.}}
:$\eqclass {a_1 \cdot c_1 + b_1 \cdot d_1, a_1 \cdot d_1 + b_1 \cdot c_1} {} = \eqclass {a_2 \cdot c_2 + b_2 \cdot d_2, a_2 \cdot d_2 + b_2 \cdot c_2} {}$
So, by the definition of integer multiplication, this leads to:
:$\eqclass {a_1, b_1} {} \otimes \eqclass {c_1, d_1} {} = \eqclass {a_2, b_2} {} \otimes \eqclass {c_2, d_2} {}$
Thus integer multiplication has been shown to be well-defined.
{{Qed}}
{{WIP|This progresses the theory for direct implementation of cross-relation on $\N \times \N$. Needs to be linked to the general approach, which is instantiated in the existing analysis of the inverse completion. The latter may be the most general approach.}}