# Integer Addition is Well-Defined/Proof 1

## Theorem

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 $\boxtimes$.

The operation $\oplus$ on these equivalence classes is well-defined, in the sense that:

 $\ds \eqclass {a_1, b_1} {}$ $=$ $\ds \eqclass {a_2, b_2} {}$ $\ds \eqclass {c_1, d_1} {}$ $=$ $\ds \eqclass {c_2, d_2} {}$ $\ds \leadsto \ \$ $\ds \eqclass {a_1, b_1} {} \oplus \eqclass {c_1, d_1} {}$ $=$ $\ds \eqclass {a_2, b_2} {} \oplus \eqclass {c_2, d_2} {}$

## 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:

 $\ds \eqclass {a_1, b_1} {}$ $=$ $\ds \eqclass {a_2, b_2} {}$ Definition of Operation Induced by Direct Product $\, \ds \land \,$ $\ds \eqclass {c_1, d_1} {}$ $=$ $\ds \eqclass {c_2, d_2} {}$ Definition of Operation Induced by Direct Product $\ds \leadstoandfrom \ \$ $\ds a_1 + b_2$ $=$ $\ds a_2 + b_1$ Definition of Cross-Relation $\, \ds \land \,$ $\ds c_1 + d_2$ $=$ $\ds c_2 + d_1$ Definition of Cross-Relation

Then we have:

 $\ds \tuple {a_1 + c_1} + \tuple {b_2 + d_2}$ $=$ $\ds \tuple {a_1 + b_2} + \tuple {c_1 + d_2}$ Commutativity and Associativity of $+$ $\ds$ $=$ $\ds \tuple {a_2 + b_1} + \tuple {c_2 + d_1}$ from above: $a_1 + b_2 = a_2 + b_1, c_1 + d_2 = c_2 + d_1$ $\ds$ $=$ $\ds \tuple {a_2 + c_2} + \tuple {b_1 + d_1}$ Commutativity and associativity of $+$ $\ds \leadsto \ \$ $\ds \tuple {a_1 + c_1, b_1 + d_1}$ $\boxtimes$ $\ds \tuple {a_2 + c_2, b_2 + d_2}$ Definition of $\boxtimes$ $\ds \leadsto \ \$ $\ds \tuple {\tuple {a_1, b_1} \oplus \tuple {c_1, d_1} }$ $\boxtimes$ $\ds \tuple {\tuple {a_2, b_2} \oplus \tuple {c_2, d_2} }$ Definition of $\oplus$

$\blacksquare$